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Wiley Series in Microwave and Optical Engineering • Kai Chang, Series Editor 



AF inpol O- 



AF 
driver 



RF 
carrier 



Multiplier 










AA 



RF AND MICROWAVE 
TRANSMITTER DESIGN 




RF AND MICROWAVE 
TRANSMITTER DESIGN 



WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING 



KAI CHANG, Editor 

Texas A&M University 



A complete list of the titles in this series appears at the end of this volume. 



RF AND MICROWAVE 
TRANSMITTER DESIGN 



Andrei Grebennikov 

Bell Labs, Alcatel-Lucent, Ireland 




WILEY 

A JOHN WILEY & SONS, INC., PUBLICATION 



Copyright © 201 1 by John Wiley & Sons, Inc. All rights reserved. 

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Library of Congress Cataloging-in-Publication Data: 

Grebennikov, Andrei 

RF and microwave transmitter design/ Andrei Grebennikov. 
p. cm. 

Includes bibliographical references and index. 
ISBN 978-0-470-52099-4 (cloth) 

Printed in Singapore 

oBook ISBN: 978-0-470-92930-8 
ePDF ISBN: 978-0-470-92929-1 

10 987654321 



CONTENTS 



Preface 

Introduction 

References 6 

1 Passive Elements and Circuit Theory 

1 . 1 Immittance Two-Port Network Parameters 9 

1.2 Scattering Parameters 13 

1.3 Interconnections of Two-Port Networks 17 

1 .4 Practical Two-Port Networks 20 

1 .4. 1 Single-Element Networks 20 

1.4.2 7T- and T -Type Networks 21 

1 .5 Three-Port Network with Common Terminal 

1.6 Lumped Elements 26 

1 .6. 1 Inductors 26 

1 .6.2 Capacitors 29 

1 .7 Transmission Line 3 1 

1.8 Types of Transmission Lines 35 

1.8.1 Coaxial Line 35 

1.8.2 Stripline 36 

1.8.3 Microstrip Line 39 

1.8.4 Slotline 41 

1.8.5 Coplanar Waveguide 42 

1.9 Noise 44 

1 .9. 1 Noise Sources 44 

1.9.2 Noise Figure 46 

1.9.3 Flicker Noise 53 
References 53 

2 Active Devices and Modeling 

2.1 Diodes 57 

2.1.1 Operation Principle 57 

2.1.2 Schottky Diodes 59 

2.1.3 p-i-n Diodes 61 

2.1.4 Zener Diodes 62 

2.2 Varactors 63 

2.2. 1 Varactor Modeling 63 

2.2.2 MOS Varactor 65 



vi CONTENTS 



2.3 MOSFETs 70 

2.3.1 Small-Signal Equivalent Circuit 70 

2.3.2 Nonlinear /-V Models 73 

2.3.3 Nonlinear C-V Models 75 

2.3.4 Charge Conservation 78 

2.3.5 Gate-Source Resistance 79 

2.3.6 Temperature Dependence 79 

2.3.7 Noise Model 81 

2.4 MESFETs and HEMTs 83 

2.4. 1 Small-Signal Equivalent Circuit 83 

2.4.2 Determination of Equivalent Circuit Elements 85 

2.4.3 Curtice Quadratic Nonlinear Model 88 

2.4.4 Parker-Skellern Nonlinear Model 89 

2.4.5 Chalmers (Angelov) Nonlinear Model 91 

2.4.6 IAF (Berroth) Nonlinear Model 93 

2.4.7 Noise Model 94 

2.5 BJTs and HBTs 97 

2.5.1 Small-Signal Equivalent Circuit 97 

2.5.2 Determination of Equivalent Circuit Elements 98 

2.5.3 Equivalence of Intrinsic it- and T -Type Topologies 100 

2.5.4 Nonlinear Bipolar Device Modeling 102 

2.5.5 Noise Model 105 
References 107 

3 Impedance Matching 113 

3.1 Main Principles 113 

3.2 Smith Chart 116 

3.3 Matching with Lumped Elements 120 

3.3.1 Analytic Design Technique 120 

3.3.2 Bipolar UHF Power Amplifier 131 

3.3.3 MOSFET VHF High-Power Amplifier 135 

3.4 Matching with Transmission Lines 138 

3.4.1 Analytic Design Technique 138 

3.4.2 Equivalence Between Circuits with Lumped and Distributed 
Parameters 144 

3.4.3 Narrowband Microwave Power Amplifier 147 

3.4.4 Broadband UHF High-Power Amplifier 149 

3.5 Matching Networks with Mixed Lumped and Distributed Elements 151 
References 153 

4 Power Transformers, Combiners, and Couplers 155 

4.1 Basic Properties 155 

4.1.1 Three-Port Networks 155 

4.1.2 Four-Port Networks 156 

4.2 Transmission-Line Transformers and Combiners 158 



4.3 Baluns 168 

4.4 Wilkinson Power Dividers/Combiners 174 

4.5 Microwave Hybrids 182 

4.6 Coupled-Line Directional Couplers 192 
References 197 

Filters 

5.1 Types of Filters 201 

5.2 Filter Design Using Image Parameter Method 205 

5.2.1 Constant-^ Filter Sections 205 

5.2.2 »7-Derived Filter Sections 207 

5.3 Filter Design Using Insertion Loss Method 210 

5.3.1 Maximally Flat Low-Pass Filter 2 1 0 

5.3.2 Equal-Ripple Low-Pass Filter 213 

5.3.3 Elliptic Function Low-Pass Filter 216 

5.3.4 Maximally Flat Group-Delay Low-Pass Filter 219 

5.4 Bandpass and Bandstop Transformation 222 

5.5 Transmission-Line Low-Pass Filter Implementation 225 

5.5.1 Richards's Transformation 225 

5.5.2 Kuroda Identities 226 

5.5.3 Design Example 228 

5.6 Coupled-Line Filters 228 

5.6.1 Impedance and Admittance Inverters 228 

5.6.2 Coupled-Line Section 231 

5.6.3 Parallel-Coupled Bandpass Filters Using Half- Wavelength 
Resonators 234 

5.6.4 Interdigital, Combline, and Hairpin Bandpass Filters 236 

5.6.5 Microstrip Filters with Unequal Phase Velocities 239 

5.6.6 Bandpass and Bandstop Filters Using Quarter- Wavelength 
Resonators 241 

5.7 SAW and BAW Filters 243 
References 250 

Modulation and Modulators 

6.1 Amplitude Modulation 255 

6.1.1 Basic Principle 255 

6.1.2 Amplitude Modulators 259 

6.2 Single-Sideband Modulation 262 

6.2.1 Double-Sideband Modulation 262 

6.2.2 Single-Sideband Generation 265 

6.2.3 Single-Sideband Modulator 266 

6.3 Frequency Modulation 267 

6.3.1 Basic Principle 268 

6.3.2 Frequency Modulators 273 

6.4 Phase Modulation 278 



viii CONTENTS 



6.5 Digital Modulation 283 

6.5.1 Amplitude Shift Keying 284 

6.5.2 Frequency Shift Keying 287 

6.5.3 Phase Shift Keying 289 

6.5.4 Minimum Shift Keying 296 

6.5.5 Quadrature Amplitude Modulation 299 

6.5.6 Pulse Code Modulation 300 

6.6 Class-S Modulator 302 

6.7 Multiple Access Techniques 304 

6.7.1 Time and Frequency Division Multiplexing 304 

6.7.2 Frequency Division Multiple Access 305 

6.7.3 Time Division Multiple Access 305 

6.7.4 Code Division Multiple Access 306 
References 308 



7 Mixers and Multipliers 311 

7.1 Basic Theory 311 

7.2 Single-Diode Mixers 313 

7.3 Balanced Diode Mixers 318 

7.3.1 Single-Balanced Mixers 318 

7.3.2 Double-Balanced Mixers 321 

7.4 Transistor Mixers 326 

7.5 Dual-Gate FET Mixer 329 

7.6 Balanced Transistor Mixers 331 

7.6.1 Single-Balanced Mixers 331 

7.6.2 Double-Balanced Mixers 334 

7.7 Frequency Multipliers 338 
References 344 



8 Oscillators 347 

8.1 Oscillator Operation Principles 347 

8.1.1 Steady-State Operation Mode 347 

8.1.2 Start-Up Conditions 349 

8.2 Oscillator Configurations and Historical Aspect 353 

8.3 Self-Bias Condition 358 

8.4 Parallel Feedback Oscillator 362 

8.5 Series Feedback Oscillator 365 

8.6 Push-Push Oscillators 368 

8.7 Stability of Self-Oscillations 372 

8.8 Optimum Design Techniques 376 

8.8.1 Empirical Approach 376 

8.8.2 Analytic Approach 379 

8.9 Noise in Oscillators 385 

8.9.1 Parallel Feedback Oscillator 386 



CONTENTS ix 



8.9.2 Negative Resistance Oscillator 392 

8.9.3 Colpitts Oscillator 394 

8.9.4 Impulse Response Model 397 

8.10 Voltage-Controlled Oscillators 407 

8.11 Crystal Oscillators 4 1 7 

8.12 Dielectric Resonator Oscillators 423 
References 428 



9 Phase-Locked Loops 433 

9.1 Basic Loop Structure 433 

9.2 Analog Phase-Locked Loops 435 

9.3 Charge-Pump Phase-Locked Loops 439 

9.4 Digital Phase-Locked Loops 441 

9.5 Loop Components 444 

9.5.1 Phase Detector 444 

9.5.2 Loop Filter 449 

9.5.3 Frequency Divider 454 

9.5.4 Voltage-Controlled Oscillator 457 

9.6 Loop Parameters 461 

9.6.1 Lock Range 461 

9.6.2 Stability 462 

9.6.3 Transient Response 463 

9.6.4 Noise 465 

9.7 Phase Modulation Using Phase-Locked Loops 466 

9.8 Frequency Synthesizers 469 

9.8.1 Direct Analog Synthesizers 469 

9.8.2 Integer-N Synthesizers Using PLL 469 

9.8.3 Fractional-N Synthesizers Using PLL 471 

9.8.4 Direct Digital Synthesizers 473 
References 474 



10 Power Amplifier Design Fundamentals 477 

10.1 Power Gain and Stability 477 

10.2 Basic Classes of Operation: A, AB, B, and C 487 

10.3 Linearity 496 

10.4 Nonlinear Effect of Collector Capacitance 503 

10.5 DC Biasing 506 

10.6 Push-Pull Power Amplifiers 515 

10.7 Broadband Power Amplifiers 522 

10.8 Distributed Power Amplifiers 537 

10.9 Harmonic Tuning Using Load-Pull Techniques 543 
10.10 Thermal Characteristics 549 

References 552 



X CONTENTS 



11 High-Efficiency Power Amplifiers 557 

11.1 Class D 557 

11.1.1 Voltage-Switching Configurations 557 

11.1.2 Current-Switching Configurations 561 

11.1.3 Drive and Transition Time 564 

11.2 Class F 567 

11.2.1 Idealized Class F Mode 569 

1 1.2.2 Class F with Quarterwave Transmission Line 572 

11.2.3 Effect of Saturation Resistance 575 

11.2.4 Load Networks with Lumped and Distributed Parameters 577 

11.3 Inverse Class F 581 

1 1.3.1 Idealized Inverse Class F Mode 583 

1 1 .3.2 Inverse Class F with Quarterwave Transmission Line 585 

11.3.3 Load Networks with Lumped and Distributed Parameters 586 

1 1 .4 Class E with Shunt Capacitance 5 89 

11.4.1 Optimum Load Network Parameters 590 

1 1.4.2 Saturation Resistance and Switching Time 595 

11.4.3 Load Network with Transmission Lines 599 

1 1 .5 Class E with Finite dc-Feed Inductance 601 

11.5.1 General Analysis and Optimum Circuit Parameters 601 

11.5.2 Parallel-Circuit Class E 605 

11.5.3 Broadband Class E 610 

11.5.4 Power Gain 613 

1 1 .6 Class E with Quarterwave Transmission Line 615 

1 1 .6. 1 General Analysis and Optimum Circuit Parameters 615 

11.6.2 Load Network with Zero Series Reactance 622 

1 1.6.3 Matching Circuits with Lumped and Distributed Parameters 625 

11.7 Class FE 628 

1 1.8 CAD Design Example: 1.75 GHz HBT Class E MMIC Power 
Amplifier 638 

References 653 



12 Linearization and Efficiency Enhancement Techniques 657 

12.1 Feedforward Amplifier Architecture 657 

12.2 Cross Cancellation Technique 663 

12.3 Reflect Forward Linearization Amplifier 665 

12.4 Predistortion Linearization 666 

12.5 Feedback Linearization 672 

12.6 Doherty Power Amplifier Architectures 678 

12.7 Outphasing Power Amplifiers 685 

12.8 Envelope Tracking 691 

12.9 Switched Multipath Power Amplifiers 695 

12.10 Kahn EER Technique and Digital Power Amplification 702 

12.10.1 Envelope Elimination and Restoration 702 

12.10.2 Pulse-Width Carrier Modulation 704 



CONTENTS xi 



12.10.3 Class S Amplifier 706 

12.10.4 Digital RF Amplification 706 
References 709 

13 Control Circuits 717 

13.1 Power Detector and VSWR Protection 7 1 7 

13.2 Switches 722 

13.3 Phase Shifters 728 

13.3.1 Diode Phase Shifters 729 

13.3.2 Schiffman 90° Phase Shifter 736 

13.3.3 MESFET Phase Shifters 739 

13.4 Attenuators 741 

13.5 Variable Gain Amplifiers 746 

13.6 Limiters 750 
References 753 

14 Transmitter Architectures 759 

14. 1 Amplitude-Modulated Transmitters 759 

14.1.1 Collector Modulation 760 

14.1.2 Base Modulation 762 

14.1.3 Low-Level Modulation 764 

14.1.4 Amplitude Keying 765 

14.2 Single-Sideband Transmitters 766 

14.3 Frequency-Modulated Transmitters 768 

14.4 Television Transmitters 772 

14.5 Wireless Communication Transmitters 776 

14.6 Radar Transmitters 782 

14.6.1 Phased-Array Radars 783 

14.6.2 Automotive Radars 786 

14.6.3 Electronic Warfare 791 

14.7 Satellite Transmitters 794 

14.8 Ultra- Wideband Communication Transmitters 797 
References 802 



Index 



809 



PREFACE 



The main objective of this book is to present all relevant information required to design the transmit- 
ters in general and their main components in particular in different RF and microwave applications 
including well-known historical and recent novel architectures, theoretical approaches, circuit simu- 
lation results, and practical implementation techniques. This comprehensive book can be very useful 
for lecturing to promote the systematic way of thinking with analytical calculations and practical 
verification, thus making a bridge between theory and practice of RF and microwave engineering. 
As a result, this book is intended for and can be recommended to university-level professors as a 
comprehensive material to help in lecturing for graduate and postgraduate students, to researchers 
and scientists to combine the theoretical analysis with practical design and to provide a sufficient 
basis for innovative ideas and circuit design techniques, and to practicing designers and engineers as 
the book contains numerous well-known and novel practical circuits, architectures, and theoretical 
approaches with detailed description of their operational principles and applications. 

Chapter 1 introduces the basic two-port networks describing the behavior of linear and nonlinear 
circuits. To characterize the nonlinear properties of the bipolar or field-effect transistors, their equiv- 
alent circuit elements are expressed through the impedance Z-parameters, admittance Y -parameters, 
or hybrid //-parameters. On the other hand, the transmission ABCD-parameters are very important 
for the design of the distributed circuits such as a transmission line or cascaded elements, whereas 
the scattering S-parameters are widely used to simplify the measurement procedure. The design for- 
mulas and curves are given for several types of transmission lines including stripline, microstrip line, 
slotline, and coplanar waveguide. Monolithic implementation of lumped inductors and capacitors is 
usually required at microwave frequencies and for portable devices. Knowledge of noise phenomena 
such as noise figure, additive white noise, low-frequency fluctuations, or flicker noise in active or 
passive elements is very important for the oscillator modeling in particular and entire transmitter 
design in general. 

In Chapter 2, all necessary steps to provide an accurate device modeling procedure starting with 
the determination of the device small-signal equivalent circuit parameters are described and discussed. 
A variety of nonlinear models for MOSFET, MESFET, HEMT, and BJT devices including HBTs, 
which are very prospective for modern microwave monolithic integrated circuits, are given. In order 
to highlight the advantages or drawbacks of one over another nonlinear device model, a comparison 
of the measured and modeled volt-ampere and voltage-capacitance characteristics, as well as a 
frequency range of model application, are analyzed. 

The main principles and impedance matching tools are described in Chapter 3. Generally, an 
optimum solution depends on the circuit requirements, such as the simplicity in practical realization, 
frequency bandwidth and minimum power ripple, design implementation and adjustability, stable 
operation conditions, and sufficient harmonic suppression. As a result, many types of the matching 
networks are available, including lumped elements and transmission lines. To simplify and visualize 
the matching design procedure, an analytical approach, which allows calculation of the parameters of 
the matching circuits using simple equations, and Smith chart traces are discussed. In addition, several 
examples of the narrowband and broadband power amplifiers using bipolar or MOSFET devices are 
given, including successive and detailed design considerations and explanations. 

Chapter 4 describes the basic properties of the three-port and four-port networks, as well as a 
variety of different combiners, transformers, and directional couplers for RF and microwave power 



xiii 



xiv PREFACE 



applications. For power combining in view of insufficient power performance of the active devices, it 
is best to use the coaxial-cable combiners with ferrite core to combine the output powers of RF power 
amplifiers intended for wideband applications. Since the device output impedance is usually too small 
for high power level, to match this impedance with a standard 50-Q load, it is necessary to use the 
coaxial-line transformers with specified impedance transformation. For narrowband applications, the 
N-way Wilkinson combiners are widely used due to the simplicity of their practical realization. At 
the same time, the size of the combiners should be very small at microwave frequencies. Therefore, 
the commonly used hybrid microstrip combiners including different types of the microwave hybrids 
and directional couplers are described and analyzed. 

Chapter 5 introduces the basic types of RF and microwave filters based on the low-pass or high- 
pass sections and bandpass or bandstop transformation. Classical filter design approaches using image 
parameter and insertion loss methods are given for low-pass and high-pass LC filter implementa- 
tions. The quarterwave-line and coupled-line sections, which are the basic elements of microwave 
transmission-line filters, are described and analyzed. Different examples of coupled-line filters in- 
cluding interdigital, combline, and hairpin bandpass filters are given. Special attention is paid to 
microstrip filters with unequal phase velocities, which can provide unexpected properties because 
of different implementation technologies. Finally, the typical structures, implementation technology, 
operational principles, and band performance of the filters based on surface and bulk acoustic waves 
are presented. 

Chapter 6 discusses the basic features of different types of analog modulation including amplitude, 
single-sideband, frequency, and phase modulation, and basic types of digital modulation such as 
amplitude shift keying, frequency shift keying, phase shift keying, or pulse code modulation and 
their variations. The principle of operations and various schematics of the modulators for different 
modulation schemes including Class S modulator for pulse-width modulation are described. Finally, 
the concept of time and frequency division multiplexing is introduced, as well as a brief description 
of different multiple access techniques. 

A basic theory describing the operational principles of frequency conversion in receivers and 
transmitters is given in Chapter 7. The different types of mixers, from the simplest based on a single 
diode to a balanced and double-balanced type based on both diodes and transistors, are described and 
analyzed. The special case is a mixer based on a dual-gate transistor that provides better isolation 
between signal paths and simple implementation. The frequency multipliers that historically were a 
very important part of the vacuum-tube transmitters can extend the operating frequency range. 

Chapter 8 presents the principles of oscillator design, including start-up and steady-state opera- 
tion conditions, noise and stability of oscillations, basic oscillator configurations using lumped and 
transmission-line elements, and simplified equation-based oscillator analyses and optimum design 
techniques. An immittance design approach is introduced and applied to the series and parallel feed- 
back oscillators, including circuit design and simulation aspects. Voltage-controlled oscillators and 
their varactor tuning range and linearity for different oscillator configurations are discussed. Finally, 
the basic circuits and operation principles of crystal and dielectric resonator oscillators are given. 

Chapter 9 begins with description of the basic phase-locked loop concept. Then, the basic perfor- 
mance and structures of the analog, charge-pump, and digital phase-locked loops are analyzed. The 
basic loop components such as phase detector, loop filter, frequency divider, and voltage-controlled 
oscillator are discussed, as well as loop dynamic parameters. The possibility and particular realiza- 
tions of the phase modulation using phase-locked loops are presented. Finally, general classes of 
frequency synthesizer techniques such as direct analog synthesis, indirect synthesis, and direct digital 
synthesis are discussed. The proper choice of the synthesizer type is based on the number of fre- 
quencies, frequency spacing, frequency switching time, noise, spurious level, particular technology, 
and cost. 

Chapter 10 introduces the fundamentals of the power amplifier design, which is generally a com- 
plicated procedure when it is necessary to provide simultaneously accurate active device modeling, 
effective impedance matching depending on the technical requirements and operation conditions, 
stability in operation, and simplicity in practical implementation. The quality of the power amplifier 



PREFACE XV 



design is evaluated by realized maximum power gain under stable operation condition with minimum 
amplifier stages, and the requirement of linearity or high efficiency can be considered where it is 
needed. For a stable operation, it is necessary to evaluate the operating frequency domains where the 
active device may be potentially unstable. To avoid the parasitic oscillations, the stabilization circuit 
technique for different frequency domains (from low frequencies up to high frequencies close to the 
device transition frequency) is discussed. The key parameter of the power amplifier is its linearity, 
which is very important for many wireless communication applications. The relationships between 
the output power, 1-dB gain compression point, third-order intercept point, and intermodulation dis- 
tortions of the third and higher orders are given and illustrated for different active devices. The device 
bias conditions, which are generally different for linearity or efficiency improvement, depend on the 
power amplifier operation class and the type of the active device. The basic Classes A, AB, B, and C 
of the power amplifier operation are introduced, analyzed, and illustrated. The principles and design 
of the push-pull amplifiers using balanced transistors, as well as broadband and distributed power 
amplifiers, are discussed. Harmonic-control techniques for designing microwave power amplifiers 
are given with description of a systematic procedure of multiharmonic load-pull simulation using the 
harmonic balance method and active load-pull measurement system. Finally, the concept of thermal 
resistance is introduced and heatsink design issues are discussed. 

Modern commercial and military communication systems require the high-efficiency long-term 
operating conditions. Chapter 1 1 describes in detail the possible circuit solutions to provide a high- 
efficiency power amplifier operation based on using Class D, Class F, Class E, or their newly 
developed subclasses depending on the technical requirements. In all cases, an efficiency improvement 
in practical implementation is achieved by providing the nonlinear operation conditions when an 
active device can simultaneously operate in pinch-off, active, and saturation regions, resulting in 
nonsinusoidal collector current and voltage waveforms, symmetrical for Class D and Class F and 
asymmetrical for Class E (DE, FE) operation modes. In Class F amplifiers analyzed in frequency 
domain, the fundamental-frequency and harmonic load impedances are optimized by short-circuit 
termination and open-circuit peaking to control the voltage and current waveforms at the device 
output to obtain maximum efficiency. In Class E amplifiers analyzed in time domain, an efficiency 
improvement is achieved by realizing the on/off active device switching operation (the pinch-off and 
saturation modes) with special current and voltage waveforms so that high voltage and high current 
do not concur at the same time. 

In modern wireless communication systems, it is very important to realize both high-efficiency 
and linear operation of the power amplifiers. Chapter 12 describes a variety of techniques and 
approaches that can improve the power amplifier performance. To increase efficiency over power 
backoff range, the Doherty, outphasing, and envelope-tracking power amplifier architectures, as well 
as switched multipath power amplifier configurations, are discussed and analyzed. There are several 
linearization techniques that provide linearization of both entire transmitter system and individual 
power amplifier. Feedforward, cross cancellation, or reflect forward linearization techniques are 
available technologies for satellite and cellular base station applications achieving very high linearity 
levels. The practical realization of these techniques is quite complicated and very sensitive to both 
the feedback loop imbalance and the parameters of its individual components. Analog predistortion 
linearization technique is the simplest form of power amplifier linearization and can be used for 
handset application, although significant linearity improvement is difficult to realize. Different types 
of the feedback linearization approaches, together with digital linearization techniques, are very 
attractive to be used in handset or base station transmitters. Finally, the potential semidigital and 
digital amplification approaches are discussed with their architectural advantages and problems in 
practical implementation. 

Chapter 13 discusses the circuit schematics and main properties of the semiconductor control 
circuits that are usually characterized by small size, low power consumption, high-speed performance, 
and operating life. Generally, they can be built based on the p-i-n diodes, silicon MOSFET, or GaAs 
MESFET transistors and can be divided into two basic parts: amplitude and phase control circuits. 
The control circuits are necessary to protect high power devices from excessive peak voltage or 



xvi PREFACE 



dc current conditions. They are also used as switching elements for directing signal between different 
transmitting paths, as variable gain amplifiers to stabilize transmitter output power, as attenuators and 
phase shifters to change the amplitude and phase of the transmitting signal paths in array systems, or 
as limiters to protect power-sensitive components. 

Finally, Chapter 14 describes the different types of radio transmitter architectures, history of radio 
communication, conventional types of radio transmission, and modern communication systems. 
Amplitude-modulated transmitters representing the oldest technique for radio communication are 
based on high- or low-level modulation methods, with particular case of an amplitude keying. 
Single-sideband transmitters as the next-generation transmitters could provide higher efficiency due 
to the transmission of a single sideband only. Frequency-modulated transmitters then became a 
revolutionary step to improve the quality of a broadcast transmission. TV transmitters include different 
modulation techniques for transmitting audio and video information, both analog and digital. Wireless 
communication transmitters as a part of the cellular technologies provide a worldwide wireless radio 
access. Radar transmitters are required for many commercial and military applications such as 
phased-array radars, automotive radars, or electronic warfare systems. Satellite transmission systems 
contribute to worldwide transmission of any communication signals through satellite transponders and 
offer communication for areas with any population density and location. Ultra-wideband transmission 
is very attractive for their low-cost and low-power communication applications, occupying a very 
wide frequency range. 



ACKNOWLEDGMENTS 

To Drs. Frederick Raab and Lin Fujiang for useful comments and suggestions in book organization 
and content covering. 

To Dr. Frank Mullany from Bell Labs, Ireland, for encouragement and support. 

The author especially wishes to thank his wife, Galina Grebennikova, for performing computer- 
artwork design, as well as for her constant support, inspiration, and assistance. 



Andrei Grebennikov 



INTRODUCTION 



A vacuum-tube or solid-state radio transmitter is essentially a source of a radio-frequency (RF) 
signal to be transmitted through antenna in different radio systems such as wireless communication, 
television (TV) and broadcasting, navigation, radar, or satellite, the information format and electrical 
performance of which should satisfy the corresponding standard requirements. The transmission 
of radio signals is produced by modulation of different types, with different output power and 
transmission mode, and in different frequency ranges, from high frequencies to millimeter waves. 
Transmitters in which the power output is generated directly by the modulated source are considered 
as possessing high-level modulation systems. In contrast, arrangements in which the modulation takes 
place at a power level less than the transmitter output are referred to as low-level modulation systems. 

Figure 1.1 shows the simplified block diagram of a conventional radio transmitter intended to 
operate at radio and microwave frequencies, which consists of the following: the source of the 
information signal, which is usually amplified, filtered, or transformed to the intermediate frequency; 
the local oscillator and frequency multiplier, which establish the stabilized carrier frequency or 
some multiple of it; the RF modulator or mixer, which combines the signal and carrier frequency 
components to produce one of the varieties of the RF modulated waves; the power amplifier to deliver 
the RF modulated signal of required power level to the antenna; the antenna duplexer to separate 
and isolate transmitting and receiving paths. The power amplifier usually consists of cascaded gain 
stages, and each stage should have adequate linearity in the case of transmitting signal with variable 
amplitude corresponding to amplitude-modulated or multicarrier signals. In practice, there are many 
variations in transmitter architectures depending on the particular frequency bandwidth, output power, 
or modulation scheme. 

Dr. Lee De Forest was an inventor who changed the world with electronics by inventing the 
vacuum tube, which he called the audion. In January 1907, De Forest filed a patent for an oscillation 
detector based on a three-electrode device representing a vacuum tube [1]. His pioneering innovation 
was the insertion of a third electrode (grid) in between the cathode (filament) and the anode (plate) of 
the previously invented diode. However, it was not until 1911 that De Forest built the first vacuum- 
tube amplifier based on three audions as amplifiers [2]. The original audion was capable of slightly 
amplifying received signals, but at this stage could not be used for more advanced applications such as 
radio transmitters. The inefficient design of the original audion meant that it was initially of little value 
to radio, and due to its high cost and short life it was rarely used. Eventually, vacuum-tube design was 
improved enough to make vacuum tubes more than novelties. Beginning in 1912, various researchers 
discovered that properly constructed (i.e., according to scientific and engineering principles) vacuum 
tubes could be employed in electrical circuits that made radio receivers and amplifiers thousands of 
times more powerful, and could also be used to make compact and efficient radio transmitters, which 
for the first time made radio broadcasting practical. 

In 1914, the first vacuum-tube radio transmitters began to appear — a key technical development 
that would lead to the introduction of widespread broadcasting. Both amateurs and commercial firms 
started to experiment with the new vacuum-tube transmitters, employing them for a variety of pur- 
poses. Six years after suspending his efforts to make audio transmissions, when he had unsuccessfully 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



1 



2 INTRODUCTION 



Signal 
source 



Local 
oscillator 



Modulator/ 
mixer 



Frequency 
multiplier 



Power 
amplifier 



Antenna 
duplexer 



Rx 



FIGURE 1.1 Conventional transmitter architecture. 



tried to use arc-transmitters, De Forest again took up developing radio to transmit sounds, including 
broadcasting news and entertainment, this time with much more success. He demonstrated that with 
this form of transmitter it was possible to telephone one to three miles, and by means of the small 
3.5-V amplifier tube used with this apparatus, direct current could be transformed to alternating 
current at frequencies from 60 cycles per second to 1,000,000 cycles per second [3], It is interesting 
that De Forest recognized the irony that he had overlooked the potential of developing his audion 
as a radio transmitter at the beginning. Reviewing his earlier arc-transmitter efforts, he wrote in his 
autobiography that he had been "totally unaware of the fact that in the little audion tube, which I was 
then using only as a radio detector, lay dormant the principle of oscillation which, had I but realized it, 
would have caused me to unceremoniously dump into the ash can all of the fine arc mechanisms which 
I had ever constructed, a procedure which a few years later actually took place all over the world" 
[4]. Meanwhile, in June 1915, the American Telephone & Telegraph Company installed a powerful 
experimental vacuum-tube transmitter in Arlington, Virginia, which quickly achieved remarkable 
distances for its audio transmissions [5]. The Marconi companies joined those experimenting with 




PHOTO 1.1 Lee De Forest. 



INTRODUCTION 3 



the new vacuum-tube transmitters, achieving an oversea working range of 50 km between ship aerials 
[6]. With the capable assistance of engineers including H. Round and C. Franklin, Guglielmo Marconi 
began experimenting with shortwave vacuum-tube transmitters by about 1916 [7]. 

However, Alexander Meissner was the first to amplify high-frequency radio signals by using a 
regenerative (positive) feedback in a vacuum triode. This principle of reactive coupling, which was 
given in its general form by Meissner in March 1913, later became the basis of the radio transmitter 
development [8]. The first high-frequency vacuum tube transmitters of small power up to 15 W were 
built by the Telefunken Company at the beginning of 1915. To provide high operating efficiency 
of the transmitter, the plate current of the special wave form, or else an auxiliary voltage of triple 
frequency, was impressed on the grid; thus greatly reducing the losses because, at the time when the 
highest voltages are applied to the tube, the passage of current through it is prevented. At the very 
beginning of 1920s, 1-kW transmitters had already been in use for radio telephony and telegraphy 
in several of the larger German cities. Much higher transmitting power was achieved by connecting 
in parallel eight or more tubes, each delivering 1.5 kW. In Russia during 1920, the 5.5-kW radio 
transmitter, where the modulated oscillations were amplified by a tube and transferred to the grids of 
six tubes in parallel that fed the antenna of 120-meter height, covered long distances of more than 
4500 km [9]. An attempt in the field of radio television had been tried out in 1920 in order to provide 
a radio transmission of photographs with two antennas: one of which sends the synchronizing signal 
while the other sends the actual picture. 

Edwin H. Armstrong is widely regarded as one of the foremost contributors to the field of 
radio engineering, being responsible for the regenerative circuit (1912), the superheterodyne circuit 
(1918), and the complete frequency-modulation radio broadcasting system (1933). Armstrong studied 
the audion for several years, performed extensive measurements, and understood and explained its 
operation when he devised a circuit, in which part of the current at the plate was fed back to the grid 
to strengthen the incoming signal. This discovery led to the independent invention of regeneration (or 
feedback) principle and the vacuum-tube oscillator. He then disproved the currently accepted theory 




PHOTO 1.2 Alexander Meissner. 



4 INTRODUCTION 




PHOTO 1.3 Edwin Armstrong. 



of the action of the triode (three-electrode vacuum tube), and published the correct explanation in 
1914 [10]. 

At the same time, the theoretical development of quantum mechanics during the 1920s played 
an important role in driving solid-state electronics, with understanding of the differences between 
metals, insulators, and semiconductors [11]. These continuing theoretical efforts then quickly led to 
the discovery of new devices, when Julius Lilienfeld invented the concept of a field-effect transistor 
in 1926 [12]. He believed that applying a voltage to a poorly conducting material would change its 
conductivity and thereby achieve amplification, but no one was able to do anything practically with 
this device until much later time. In conjunction with this, it is worth mentioning that the oscillating 
crystal detector was described by W. Eccles in 1909 and then practically implemented as an oscillator 
and even a low-power transmitter (based on one-port negative resistance principle) by O. Losev in 
early 1920s [13,14]. All details needed to duplicate these circuits to make a tunnel-diode oscillator 
were reported in the September 1924 issue of Radio News and in the 1st and 8th October 1924 issues 
of Wireless World, with predictions that crystals would someday replace valves in electronics. 

Shortly after the end of the war in 1945, Bell Labs formed a Solid State Physics Group led 
by William Shockley, with an assignment to seek a solid-state alternative to fragile glass vacuum- 
tube amplifiers. This group made a very important decision right at the beginning: that the simplest 
semiconductors were silicon and germanium and that their efforts would be directed at those two 
elements. The first attempts were based on Shockley's ideas about using an external electrical 
field on a semiconductor to affect its conductivity, but these experiments failed every time in all 
sorts of configurations and materials. The group was at a standstill until John Bardeen suggested 
a theory that invoked surface states that prevented the field from penetrating the semiconductor. 
In November 1947, John Bardeen and Walter Brattain, working without Shockley, succeeded in 
creating a point-contact transistor that achieved amplification when electrical field was applied to a 
crystal of germanium [15]. At the same time, Shockley secretly continued and successfully finished 
his own work to build a different sort of transistor based on n-p junctions that depended on the 
introduction of the minority carriers instead of point contacts, which he expected would be more 



INTRODUCTION 5 




PHOTO 1.4 William Shockley. 



likely to be commercially viable [16]. In his seminal work Electrons and Holes in Semiconductors 
with Applications to Transistor Electronics (1955), Shockley worked out the critical ideas of drift 
and diffusion and the differential equations that govern the flow of electrons in solid-state crystals, 
where the Shockley ideal diode equation was also described. The term "transistor" was coined by 
John Pierce, who later recalled: ". . . at that time, it was supposed to be the dual of the vacuum tube. 
The vacuum tube had transconductance, so the transistor would have transresistance. And the name 
should fit in with the names of other devices, such as varistor and thermistor. And ... I suggested the 
name transistor" 

The first to perceive the possibility of integrated circuits based upon semiconductor technology was 
Geoffrey Dummer, who said addressing the Electronic Components Conference in 1952: "With the 
advent of the transistor and the work in semiconductors generally, it seems now possible to envisage 
electronic equipment in a solid block with no connecting wires. The block may consist of layers of 
insulating, conducting, rectifying, and amplifying materials, the electrical functions being connected 
directly by cutting out areas of the various layers" [17]. In 1958, Jack Kilby of Texas Instruments 
developed the first integrated circuit consisting of a few mesa transistors, diffused capacitors, and bulk 
resistors on a piece of germanium using gold wires for interconnections [18]. The integrated circuit 
developed by Robert Noyce of Fairchild could resolve problems with wires by adding a final metal 
layer and then taking away some of it so that the wires required for the components to be connected 
were shaped so as to make the integrated circuit more suitable for mass production [19]. A year 
later, a planar process was developed by Jean Hoerni that utilized heat diffusion process, and oxide 
passivation of the surface protected the junctions and provided a reproducibility that assured more 
consistency than any previous manufacturing process. The first microwave gallium-arsenide (GaAs) 
Schottky-gate field-effect transistor (metal semiconductor field-effect transistor or MESFET), which 
had a maximum frequency / max = 3 GHz, was reported in 1967 [20]. But it was not until 1976 that the 
first fully monolithic single-stage GaAs MESFET X-band broadband amplifier was developed based 
on lumped matching elements [21]. Eight years later, on a GaAs chip of approximately the same area, 
an entire X-band transmit-receive (T/R) module was fabricated, consisting of two switches, a four-bit 



6 INTRODUCTION 



TABLE 1.1 IEEE Standard Letter Designations for Frequency Bands. 



Band Frequency Wavelength 



HF (high frequency) 


3-30 MHz 


100-10 m 


VHF (very high frequency) 


30-300 MHz 


10-1.0 m 


UHF (ultra high frequency) 


300-1000 MHz 


1.0 m to 30 cm 


L 


1-2 GHz 


30-15 cm 


S 


2-4 GHz 


15 cm to 7.5 cm 


C 


4-8 GHz 


7.50-3.75 cm 


X 


8-12 GHz 


3.75-2.50 cm 


Ku 


12-18 GHz 


2.50-1.67 cm 


K 


18-27 GHz 


1.67-1.11 cm 


Ka 


27^10 GHz 


1.11 cm to 7.5 mm 


V 


40-75 GHz 


7.5^1.0 mm 


w 


75-1 10 GHz 


4.0-2.7 mm 


Millimeter wave 


110-300 GHz 


2.7-1.0 mm 



phase shifter, a three-stage low-noise amplifier, and a four-stage power amplifier [22]. Generally, a 
monolithic microwave integrated circuit (MMIC) was defined as an active or passive microwave circuit 
formed in situ on a semiconductor substrate by a combination of deposition techniques including 
diffusion, evaporation, epitaxy, and other means [23]. 

Table 1.1 shows the IEEE standard letter designations for frequency bands [24], In military radar 
band designations, millimeter-wave bandwidth occupies the frequency range from 40 to 300 GHz. 
The letter designations (L, S, C, X, Ku, K, Ka) were meant to be used for radar, but have become 
commonly used for other microwave frequency applications. The K-band is the middle band (18-27 
GHz) that originated from the German word "Kurz", which means short, while Ku-band is lower in 
frequency (Kurz-under), and Ka-band is higher in frequency (Kurz-above). The 1984 revision defined 
the application of letters V and W to a portion of the millimeter- wave region each while retaining the 
previous letter designators for frequencies. 

REFERENCES 

1. L. De Forest, "Space Telegraphy," U.S. Patent 879,532, Feb. 1908 (filed Jan. 1907). 

2. L. De Forest, "The Audion - Detector and Amplifier," Proc. IRE, vol. 2, pp. 15-29, Mar. 1914. 

3. "High-Frequency Oscillating Transmitter for Wireless Telephony," Electrical World, p. 144, July 18, 1914. 

4. T. H. White, United States Early Radio History, http://earlyradiohistory.us, sec. 11. 

5. E. B. Craft and E. H. Colpitts, "Radio Telephony," Trans. AIEE, vol. 38, pp. 305-343, Feb. 1919. 

6. "Wireless-Telephone Set," Electrical World, p. 487, Sept. 5, 1914. 

7. J. E. Brittain, "Electrical Engineering Hall of Fame: Guglielmo Marconi," Proc. IEEE, vol. 92, pp. 1501-1504, 
Sept. 2004. 

8. A. Meissner, "The Development of Tube Transmitters by the Telefunken Company," Proc. IRE, vol. 10, 
pp. 3-23, Jan. 1922. 

9. V. Bashenoff, "Progress in Radio Engineering in Russia 1918-1922," Proc. IRE, vol. 11, pp. 257-270, Mar. 
1923. 

10. "Edwin Howard Armstrong," Proc. IRE, vol. 31, p. 315, July 1943. 

11. W. F. Brinkman, D. E. Haggan, and W. W. Troutman, "A History of the Invention of the Transistor and Where 
It Will Lead Us," IEEE J. Solid-State Circuits, vol. SC-32, pp. 1858-1865, Dec. 1997. 

12. J. E. Lilienfeld, "Method and Apparatus for Controlled Electric Currents," U.S. Patent 1745175, Jan. 1930 
(filed Oct. 1926). 



REFERENCES 7 



13. W. H. Eccles, "On an Oscillation Detector Actuated Solely by Resistance-Temperature Variations," Proc. 
Phys. Society: London, vol. 22, pp. 360-368, Jan. 1909. 

14. M. A. Novikov, "Oleg Vladimirovich Losev: Pioneer of Semiconductor Electronics," Physics of the Solid 
State, vol. 46, pp. Jan. 2004. 

15. J. Bardin and W. H. Brattain, "Three-Electrode Circuit Element Utilizing Semiconductive Materials," U.S. 
Patent 2524035, Oct. 1950 (filed June 17, 1948). 

16. W. Shockley, "Circuit Element Utilizing Semiconductive Material," U.S. Patent 2569347, Sept. 1951 (filed 
June 26, 1948). 

17. J. S. Kilby, "Invention of the Integrated Circuit," IEEE Trans. Electron Devices, vol. ED-23, pp. 648-654, 
July 1976. 

18. J. S. Kilby, "Miniaturized Electronic Circuits," U.S. Patent 3138743, June 1964 (filed Feb. 1959). 

19. R.N. Noyce, "Semiconductor Device-and-Lead Structure," U.S. Patent 2981877,Apr. 1961 (filedjuly 1959). 

20. D. N. McQuiddy, J. W. Wassel, J. Bradner LaGrange, and W. R. Wisseman, "Monolithic Microwave Integrated 
Circuits: An Historical Perspective," IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 997-1008, Sept. 
1984. 

21. R. S. Pengelly and J. A. Turner, "Monolithic Broadband GaAs F.E.T. Amplifiers," Electronics Lett., vol. 12, 
pp. 251-252, May 1976. 

22. E. C. Niehenke, R. A. Pucel, and I. J. Bahl, "Microwave and Millimeter- Wave Integrated Circuits," IEEE 
Trans. Microwave Theory Tech., vol. MTT-50, pp. 846-857, Mar. 2002. 

23. R. A. Pucel, "Design Considerations for Monolithic Microwave Circuits," IEEE Trans. Microwave Theory 
Tech., vol. MTT-29, pp. 513-534, June 1981. 

24. IEEE Standard 521-2002, IEEE Standard Letter Designations for Radar-Frequency Bands, 2003. 




Passive Elements and Circuit 




The two-port equivalent circuits are widely used in radio frequency (RF) and microwave circuit 
design to describe the electrical behavior of both active devices and passive networks [1-4]. The 
two-port network impedance Z-parameters, admittance T-parameters, or hybrid //-parameters are 
very important to characterize the nonlinear properties of the active devices, bipolar or field-effect 
transistors. The transmission AB CD-parameters of a two-port network are very convenient for design- 
ing the distributed circuits like transmission lines or cascaded elements. The scattering 5-parameters 
are useful to characterize linear circuits, and are required to simplify the measurement procedure. 
Transmission lines are widely used in matching circuits in power amplifiers, in resonant circuits 
in the oscillators, filters, directional couplers, power combiners, and dividers. The design formulas 
and curves are presented for several types of transmission lines including stripline, microstrip line, 
slotline, and coplanar waveguide. Monolithic implementation of lumped inductors and capacitors is 
usually required at microwave frequencies and for portable devices. Knowledge of noise phenomena, 
such as the noise figure, additive white noise, low-frequency fluctuations, or flicker noise in active 
or passive elements, is very important for the oscillator modeling in particular and entire transmitter 
design in general. 



1.1 IMMITTANCE TWO-PORT NETWORK PARAMETERS 

The basic diagram of a two-port nonautonomous transmission system can be represented by the 
equivalent circuit shown in Figure 1.1, where Vs is the independent voltage source, Zs is the source 
impedance, LN is the linear time-invariant two-port network without independent source, and Z L is 
the load impedance. The two independent phasor currents / t and I 2 (flowing across input and output 
terminals) and phasor voltages V\ and V 2 characterize such a two-port network. For autonomous 
oscillator systems, in order to provide an appropriate analysis in the frequency domain of the two- 
port network in the negative one-port representation, it is sufficient to set the source impedance to 
infinity. For a power amplifier or oscillator design, the elements of the matching or resonant circuits, 
which are assumed to be linear or appropriately linearized, can be found among the /JV-network 
elements, or additional two-port linear networks can be used to describe their frequency domain 
behavior. 

For a two-port network, the following equations can be considered to be imposed boundary 
conditions: 



Suppose that it is possible to obtain a unique solution for the linear time-invariant circuit shown in 
Figure 1.1. Then the two linearly independent equations, which describe the general two-port network 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



V x + Z s h = V s 



(1.1) 



V 2 + Z L I 2 = V L . 



(1.2) 



9 



10 PASSIVE ELEMENTS AND CIRCUIT THEORY 




FIGURE 1.1 Basic diagram of two-port nonautonomous transmission system. 



in terms of circuit variables V\, V2, h, an d h, can be expressed in a matrix form as 

[M] [V] + [JV] [/] = 0 



(1.3) 



m n Vi + m l2 V 2 + «n/i + n l2 h = 0 
m 2l Vi + m 2 2V 2 + n 2i Ii + n 22 I 2 = 0 



(1.4) 



The complex 2x2 matrices [M] and [N] in Eq. (1.3) are independent of the source and load 
impedances Z s and Z L and voltages Vs and V L , respectively, and they depend only on the circuit 
elements inside the LN network. 

If matrix [M] in Eq. (1.3) is nonsingular when \M\ ^ 0, then this matrix equation can be rewritten 
in terms of [/] as 



[U] = - [M]- 1 [TV] [/] = [Z] [/] 



(1.5) 



where [Z] is the open-circuit impedance two-port network matrix. In a scalar form, matrix Eq. (1.5) 
is given by 



Vi = Z n h+Z l2 l 2 
V 2 = Z21/1 + Z 22 I 2 



(1.6) 
(1.7) 



where Zn and Z 22 are the open-circuit driving-point impedances, and Z\ 2 and Z 2 \ are the open-circuit 
transfer impedances of the two-port network. The voltage components V\ and V 2 due to the input 
current /1 can be found by setting I 2 — 0 in Eqs. (1.6) and (1.7), resulting in an open-output terminal. 
Similarly, the same voltage components Vi and V 2 are determined by setting /1 = 0 when the input 
terminal becomes open-circuited. The resulting driving-point impedances can be written as 



Z n = 



whereas the two transfer impedances are 



Z„ = 



z 2 =o 



(1.8) 



/l=0 



(1.9) 



/ 2 =o 



Dual analysis can be used to derive the short-circuit admittance matrix when the current compo- 
nents I[ and/2 are considered as outputs caused by V\ and V 2 . If matrix [N] inEq. (1.3) is nonsingular 



IMMITTANCE TWO-PORT NETWORK PARAMETERS 



11 



when \N\ ^ 0, this matrix equation can be rewritten in terms of [V] as 

[I] = -[N]- 1 [M] [V] = [Y][V] 



(1.10) 



where [Y] is the short-circuit admittance two-port network matrix. In a scalar form, matrix Eq. (1.10) 
is written as 



h = 7 u Vi+Y u V 2 
I2 = Y21V1 + Y22V2 



(1.11) 
(1.12) 



where Y n and Y 12 are the short-circuit driving-point admittances, and Y l2 and Y 2 i are the short-circuit 
transfer admittances of the two-port network. In this case, the current components Ii and I 2 due to the 
input voltage source Vj are determined by setting V2 — OinEqs. (1.11) and (1.1 2), thus creating a short 
output terminal. Similarly, the same current components / t and I 2 are determined by setting Vi — 0 
when input terminal becomes short-circuited. As a result, the two driving-point admittances are 



Y - h 



whereas the two transfer admittances are 



Y - h 

in — — 



F21 = 



V, 



Yn = 



v 2 =o 



v,=o 



Vi=0 



(1.13) 



(1.14) 



In some cases, an equivalent two-port network representation can be redefined in order to express 
the voltage source Vi and output current 7 2 in terms of the input current / t and output voltage V 2 - If 
the submatrix 



m n n l2 
m 2[ n 22 



given in Eq. (1.4) is nonsingular, then 



mu "12 
m 2l n 22 



n 2i m 22 



= [H] 



h 

V 2 



(1.15) 



where [H] is the hybrid two-port network matrix. In a scalar form, it is best to represent matrix Eq. 
(1.15) as 



Vi = huh+hnVi 

h = ^2l/l + ^22^2 



(1.16) 
(1.17) 



where hn, hn, ^21 , and /122 are the hybrid //-parameters. The voltage source V\ and current component 
I 2 are determined by setting V2 — 0 for the short output terminal in Eqs. (1.16) and (1.17) as 



hu 



Vi 



v 2 =o 



(1.18) 



V 2 =0 



where h n is the driving-point input impedance and h 2l is the forward current transfer function. 
Similarly, the input voltage source V x and output current I 2 are determined by setting /1 = 0 when 



12 



PASSIVE ELEMENTS AND CIRCUIT THEORY 



input terminal becomes open-circuited as 



fin = 



/i=o 



I 2 



(1.19) 



/ 1= o 



where hu is the reverse voltage transfer function and I122 is the driving-point output admittance. 

The transmission parameters, often used for passive device analysis, are determined for the 
independent input voltage source Vi and input current Ii in terms of the output voltage V 2 and output 
current It. In this case, if the submatrix 



m 2 i n%\ 



given in Eq. (1.4) is nonsingular, we obtain 



mil «ii 
m 2 i M21 



m i2 «i2 
m%i n 22 



V 2 
-h 



[ABCD] 



V 2 
-h 



(1.20) 



where [ABCD] is the forward transmission two-port network matrix. In a scalar form, we can write 



Vi = AV 2 - BI 2 
/j = CV 2 -DI 2 



(1.21) 
(1.22) 



where A, B, C, and D are the transmission parameters. The voltage source V\ and current component 
/1 are determined by setting I 2 — 0 for the open output terminal in Eqs. (1.21) and (1.22) as 



Vi 
1 2 



C 



/ 2 =o 



1 2 



(1.23) 



1 2=0 



where A is the reverse voltage transfer function and C is the reverse transfer admittance. Similarly, 
the input independent variables V\ and /j are determined by setting V 2 — 0 when the output terminal 
is short-circuited as 



h 



D 



v 2 =o 



(1.24) 



where B is the reverse transfer impedance and D is the reverse current transfer function. The reason a 
minus sign is associated with I 2 in Eqs. (1.20) to (1.22) is that historically, for transmission networks, 
the input signal is considered as flowing to the input port whereas the output current flowing to the 
load. The direction of the current -I 2 entering the load is shown in Figure 1.2. 

The parameters describing the same two-port network through different two-port matrices 
(impedance, admittance, hybrid, or transmission) can be cross-converted, and the elements of each 




FIGURE 1.2 Basic diagram of loaded two-port transmission system. 



SCATTERING PARAMETERS 13 



TABLE 1.1 Relationships Between Z-, Y-, H- and ABCD-Parameters. 



[Z] [F] [H] [ABCD] 



[Z] 


Zn 


Z12 


Fn 


1 12 


A// h n 


,4 


AD — BC 




Z 2 i 


Z22 


AT 


~ af 


/?22 ft 22 


C 


C 








^21 


I'll 


/721 1 


1 


D 








AF 


AF 


^ 22 ft 22 


C 


C 


U] 


z 2 


Z12 


^11 


)*I2 


1 /1 12 


D 


AD - BC 




AZ 


~ AZ 


*21 


'22 


ftll ftll 


1} 


B 




Z21 


Zn 






/i 2 i AH 


1 


A 




AZ 


AZ 






ftll ftll 


~B 


B 


I" 1 


AZ 


^ 12 


J 


1 12 


ftll ft 12 


B 


AD — BC 




Z22 


Z22 






ft 2 l ft22 


75 


D 




Z21 


1 


i'21 


AF 




1 


C 




Z22 


Z2I 


1 


Fn 




~75 


D 


[ABCD] 


Zn 


AZ 


y 2 2 


1 


A// /in 




B 




Z21 


Z27 






/l21 ft21 


C 


D 




1 


Z22 


AY 


Yn 


ft22 1 








Z27 


Z21" 


~Yll 


F21 


/?21 /?21 







matrix can be expressed by the elements of other matrices. For example, Eqs. (1.11) and (1.12) for 
the F-parameters can be easily solved for the independent input voltage source Vi and input current 
Ii as 

V i = --^V 2 + — h (1.25) 

*21 *21 

h = - v V 2 + -±I 2 . (1.26) 

F21 121 

By comparing the equivalent Eqs. (1.21) and (1.22) and Eqs. (1.25) and (1.26), the direct relation- 
ships between the elements of the transmission ABCD-matrix and admittance F-matrix are written 

as 

A 
C 

where AF = Y n Y 12 - Y n Y 2l . 

A summary of the relationships between the impedance Z-parameters, admittance F-parameters, 
hybrid //-parameters, and transmission ABCD-parameters is shown in Table 1.1, where AZ = 
Z11Z22 — Z12Z21 and AH — h\\h%2 — 12^21 - 



F22 
Yn 
AY 
F21" 



D 



1 

'F2T 
_Fn 
F 2 , 



(1.27) 
(1.28) 



1.2 SCATTERING PARAMETERS 

The concept of incident and reflected voltage and current parameters can be illustrated by the one-port 
network shown in Figure 1.3, where the network impedance Z is connected to the signal source Vs 



14 



PASSIVE ELEMENTS AND CIRCUIT THEORY 




FIGURE 1.3 Incident and reflected voltages and currents. 



with the internal impedance Z s . In a common case, the terminal current / and voltage V consist 
of incident and reflected components (assume their root mean square [rms] values). When the load 
impedance Z is equal to the conjugate of source impedance expressed as Z = Z|, the terminal current 
becomes the incident current. It is calculated from 



Z* + Z s 2ReZ s 

The terminal voltage, defined as the incident voltage, can be determined from 



(1.29) 



Vt = 



Z* + Z s 



2ReZ s 



(1.30) 



Consequently, the incident power, which is equal to the maximum available power from the source, 
can be obtained by 



Pi = Re (VJ*) = 



l^sl 2 



4ReZ s 

The incident power can be presented in a normalized form using Eq. (1.30) as 



(1.31) 



UTReZs 

lz*l 2 ' 

l^sl 



(1.32) 



This allows the normalized incident voltage wave a to be defined as the square root of the incident 
power Pj by 



V.VReZs 



(1.33) 



Similarly, the normalized reflected voltage wave b, defined as the square root of the reflected 
power P t , can be written as 



VrVReZ^ 



(1.34) 



SCATTERING PARAMETERS 15 



The incident power can be expressed by the incident current 7 ; and the reflected power can be 
expressed by the reflected current 7 r , respectively, as 



Pi = |/i| 2 ReZ s 
P r = |/r| 2 ReZ s 



(1.35) 
(1.36) 



As a result, the normalized incident voltage wave a and reflected voltage wave b can be given by 

a = 7>~ = /ix/ReZs (1.37) 
b = Jp~ r = WReZs. (1.38) 

The parameters a and b can also be called the normalized incident and reflected current waves, 
or simply normalized incident and reflected waves, respectively, since the normalized current waves 
and the normalized voltage waves represent the same parameters. 

The voltage V and current /, related to the normalized incident and reflected waves a and b, can 
be written as 



V = Vi + V, 
I = h - l t = 



S a+^^b 



VReZs VReZ s 

1 ^ 1 
VReZi lVReZi 



where 



V+Z S I 
2VReZ s 



b = 



2VReZ s 



(1.39) 
(1.40) 

(1.41) 



The source impedance Zs is often purely real and, therefore, is used as the normalized impedance. 
In microwave design technique, the characteristic impedance of the passive two-port networks, 
including transmission lines and connectors, is considered as real and equal to 50 CI. This is very 
important for measuring S-parameters when all transmission lines, source, and load should have the 
same real impedance. For Z s = Z| = Z 0 , where Z 0 is the characteristic impedance, the ratio of 
the normalized reflected wave and the normalized incident wave for a one-port network is called the 
reflection coefficient T, defined as 



V 



V-Z S I 



V + Z S I V + Z S I z + z s 



(1.42) 



where Z = V /I . 

For a two-port network shown in Figure 1.4, the normalized reflected waves b\ and £>2 can also be 
represented by the normalized incident waves a\ and ai, respectively, as 



hi = S n ai + S n a 2 

hi = $2101 + S 22fl2 



(1.43) 
(1.44) 



or, in a matrix form, 



~bi~ 




~S n 






«1 


>_ 




_S 2l 






_a 2 _ 



(1.45) 



16 PASSIVE ELEMENTS AND CIRCUIT THEORY 




FIGURE 1.4 Basic diagram of S-parameter two-port network. 



where the incident waves ci\ and cii and the reflected waves b\ and £>2, for complex source and load 
impedances Z s and Z L , are given by 



Vi + z s /i y 2 + z L / 2 

Oi = . a 2 = 



2VReZ s 
2VReZ s 



2VReZ L 



2VReZ L 



(1.46) 
(1.47) 



where Sn, S12, S21, and S22 are the S-parameters of the two-port network. 
From Eq. (1.45) it follows that if ci2 — 0, then 



Sn = 



hi 



b 2 

S21 = - 

a 2 =0 °l 



a 2 =0 



(1.48) 



where Sn is the reflection coefficient and S 2 i is the transmission coefficient for ideal match- 
ing conditions at the output terminal when there is no incident power reflected from the load. 
Similarly, 



"2 



b 2 

S22 = — 

a,=0 fl 2 



(1.49) 



where Su is the transmission coefficient and S22 is the reflection coefficient for ideal matching 
conditions at the input terminal. 

To convert S-parameters to the admittance F-parameters, it is convenient to represent Eqs. (1.46) 
and (1.47) as 



h = (ai - bi) -}= h = (a 2 - b 2 ) -j== 
Vi = fa+b^JZo V l = (a 2 + b 2 )Jz 0 



(1.50) 
(1.51) 



where it is assumed that the source and load impedances are real and equal to Z s = Z L = Z 0 . 
Substituting Eqs. (1.50) and (1.51) to Eqs. (1.11) and (1.12) results in 



gi - h 
a 2 — b2 



Yn («i + h) VZo + Y l2 (a 2 + b 2 ) JZ 0 
Y 2 i (ai + h) yfZo + Y22 (a 2 + b 2 ) ^ 



(1.52) 
(1.53) 



INTERCONNECTIONS OF TWO-PORT NETWORKS 



17 



which can then be respectively converted to 

(1 + F U Z 0 ) - b 2 Y 12 Z 0 = -a x (1 - F„Z 0 ) + a 2 F 12 Z„ (1.54) 
-b x Y 2X Z a - b 2 (1 + F 22 Z 0 ) = «iF 2 iZo - a 2 (1 - F 22 Z 0 ) • (1.55) 

Eqs. (1.54) and (1.55) can be easily solved for the reflected waves b\ and b 2 as 

b x [(1 + F„Z 0 )(1 + Y 22 Z 0 ) - Y l2 Y 2l Z 2 0 ] = a, [(1 - F U Z 0 )(1 + F 22 Z 0 ) + Y l2 Y 2l Z 2 0 ] - 2a 2 Y l2 Z 0 

(1.56) 

b 2 [d + FnZo)(l + F 22 Z 0 ) - Fi 2 F 2 iZ 0 2 ] = -2 ai Y 21 Z 0 + a 2 [(1 + F n Z 0 )(l - F 22 Z 0 ) + F 12 F 21 Z 0 2 ] 

(1.57) 

Comparing equivalent Eqs. (1.43) and (1.44) and Eqs. (1.56) and (1.57) gives the following 
relationships between the scattering S-parameters and admittance F-parameters: 

s = (1 - FnZoKl + F 22 Z 0 ) + F 12 F 21 Z 0 2 
11 (l + F n Zo)(l + F 22 Z 0 )-F 12 F 21 Z 2 

-2F 12 Z 0 

Sn = — = (1.59) 

(1 + F U Z 0 )(1 + F 22 Z 0 )-F 12 F 21 Z 2 

-2F 2 ,Zo 

Sn = „ ° - - „ 2 (1-60) 



= , (1.61) 

(l + F 11 Zo)(l + F 22 Z 0 )-F 12 F 21 Z 2 

Similarly, the relationships of S-parameters with Z-, H-, and ABCD-parameters can be obtained 
for the simplified case when the source impedance Zs and the load impedance Zl are equal to the 
characteristic impedance Z 0 [5]. 



1.3 INTERCONNECTIONS OF TWO-PORT NETWORKS 

When analyzing the behavior of a particular electrical circuit in terms of the two-port network 
parameters, it is often necessary to define the parameters of a combination of the two or more internal 
two-port networks. For example, the general feedback amplifier circuit consists of an active two-port 
network representing the amplifier stage, which is connected in parallel with a passive feedback 
two-port network. In general, the two-port networks can be interconnected using parallel, series, 
series-parallel, or cascade connections. 

To characterize the resulting two-port networks, it is necessary to take into account which currents 
and voltages are common for individual two-port networks. The most convenient set of parameters 
is one for which the common currents and voltages represent a simple linear combination of the 
independent variables. For the interconnection shown in Figure 1.5(a), the two-port networks Z a and 
Z b are connected in series for both the input and output terminals. Therefore, the currents flowing 
through these terminals are equal when 



h = ha — I ib h — ha = hb 



(1.62) 




(d) 

FIGURE 1.5 Different interconnections of two-port networks. 



INTERCONNECTIONS OF TWO-PORT NETWORKS 



19 



or, in a matrix form, 



[/] = [/J = [/„] . 



(1.63) 



The terminal voltages of the resulting two-port network represent the corresponding sums of the 
terminal voltages of the individual two-port networks when 



Vi = V u + V lb V 2 = V 2d + V 2l 



(1.64) 



or, in a matrix form, 



[V] = [VJ + [V b ] ■ 



(1.65) 



The currents are common components, both for the resulting and individual two-port networks. 
Consequently, to describe the properties of such a circuit, it is most convenient to use the impedance 
matrices. For each two-port network Z a and Z b , we can write using Eq. (1.62), respectively, 



[VJ = [ZJ [/J = [ZJ [/] 
[V b ] = [Z b ] [/„] = [Z b ] [/] . 



(1.66) 
(1.67) 



Adding both sides of Eqs. (1.66) and (1.67) yields 

[V] = [Z] [/] 

where 



[Z] = [ZJ + [Z b ] 



Zlla + Znb Zi2a + Zi2b 
Z21a + Z21b Z 2 2a + Z22b 



(1.68) 



(1.69) 



The circuit shown in Figure 1.5(b) is composed of the two-port networks F a and Fb connected 
in parallel, where the common components for both resulting and individual two-port networks are 
input and output voltages, respectively, 



V 1 = V u =V a V 2 = V 2a = V 2b 



(1.70) 



or, in a matrix form, 



iv] = [yj = iv b ] . 



(1.71) 



Consequently, to describe the circuit properties in this case, it is convenient to use the admittance 
matrices that give the resulting matrix equation in the form 



[I] = [Y][V] 



(1.72) 



where 



[Y] = [Fa] + [lb] 



Flla + Yub Fl2a + Fl2b 
F21a + Y 2 u, F 2 2a + F 2 2b 



(1.73) 



The series connection of the input terminals and parallel connection of the output terminals are 
characterized by the circuit in Figure 1.5(c), which shows a series-parallel connection of two-port 



20 PASSIVE ELEMENTS AND CIRCUIT THEORY 



networks. The common components for this circuit are the input currents and the output voltages. As 
a result, it is most convenient to analyze the circuit properties using hybrid matrices. The resulting 
two-port hybrid matrix is equal to the sum of the two individual hybrid matrices written as 



[H] = [flj + [H b ] = 



hlli + hub /2l2a+/'l2b 
&21a + ^21b ^22a + ^22b 



(1.74) 



Figure 1.5(d) shows the cascade connection of the two individual two-port networks. For such an 
approach using the one-by-one interconnection of the two-port networks, the output voltage and the 
output current of the first network is equal to the input voltage and the input current of the second 
one, respectively, when 

V, = V u h = / la (1.75) 
V 2a = Vi -7 2a = /ib (1.76) 
V 2b = V 2 -I 2b = -h (1.77) 



In this case, it is convenient to use a system of ABCZ)-parameters given by Eqs. (1.21) and (1.22). 
As a result, for the first individual two-port network shown in Figure 1.5(d), 



"v la ~ 




"A a 


Sa" 




" v 2a " 






_C a 






~ 7 2a 



(1.78) 



or, using Eqs. (1.75) and (1.76), 



"vr 




"A a 


5a" 




~v lb ' 


_h_ 




_C a 






. 7 lb_ 



(1.79) 



Similarly, for the second individual two-port network, 



>lb" 




"A b 


B b ~ 




" V 2b " 




"A b 


fib" 




" v 2 ' 


. 7 lb. 




c b 


D b _ 








c b 


D b 







(1.80) 



Then, substituting matrix Eq. (1.80) to matrix Eq. (1.79) yields 







Aa Sa 




A b 


B b 




v 2 




A 


B 




h 




C a £>a 




c b 


D b 




-h 




C 


D 





(1.81) 



Consequently, the transmission matrix of the resulting two-port network obtained by the cascade 
connection of the two or more individual two-port networks is determined by multiplying the trans- 
mission matrices of the individual networks. This important property is widely used in the analysis 
and design of transmission networks and systems. 



1.4 PRACTICAL TWO-PORT NETWORKS 
1.4.1 Single-Element Networks 

The simplest networks, which include only one element, can be constructed by a series-connected 
admittance Y, as shown in Figure 1.6(a), or by a parallel-connected impedance Z, as shown in 
Figure 1.6(b). 



PRACTICAL TWO-PORT NETWORKS 



21 





(a) (P) 

FIGURE 1.6 Single-element networks. 



The two-port network consisting of the single series admittance Y can be described in a system of 
F-parameters as 



/j = YVi - YV 2 

i 2 = -y + yv 2 



(1.82) 
(1.83) 



or, in a matrix form, 



[Y] 



Y -Y 
-Y Y 



(1.84) 



which means that Y n — Y 2 2 — Y and F 12 = Y 2 i — —Y. The resulting matrix is a singular matrix with 
\Y\ — 0. Consequently, it is impossible to determine such a two-port network with the series admittance 
F-parameters through a system of Z-parameters. However, by using H- and AS CD-parameters, it can 
be described, respectively, by 



[H] 



l/Y 
-1 



[ABCD] 



l/Y 
1 



(1.85) 



Similarly, for a two-port network with the single parallel impedance Z, 

[Z] = 



z z 
z z 



(1.86) 



which means that Z n — Z 12 = Z 2i — Z 22 — Z. The resulting matrix is a singular matrix with |Z| 
0 In this case, it is impossible to determine such a two-port network with the parallel impedance 
Z-parameters through a system of F-parameters. By using H- and ABCD -parameters, this two-port 
network can be described by 



[H] 



0 1 
-1 1/Z 



[ABCD] 



1 0 
1/Z 1 



(1.87) 



1.4.2 7i- and T-Type Networks 

The basic configurations of a two-port network that usually describe the electrical properties of the 
active devices can be represented in the form of a it -circuit shown in Figure 1.7(a) and in the form 
of a T-circuit shown in Figure 1.1(b). Here, the 7r-circuit includes the current source g m Vi and the 
T-circuit includes the voltage source r m Ii. 



22 



PASSIVE ELEMENTS AND CIRCUIT THEORY 




^X^=K3 — o 



FIGURE 1.7 Basic diagrams of it- and T-networks. 



By writing the two loop equations using Kirchhoff 's current law or applying Eqs. (1.13) and (1.14) 
for the it -circuit, we obtain 



h-(Y l + Y 3 )V l + Y 3 V 2 = 0 
h + (gm ~ Y 3 ) Vi + (Y 2 + Y 3 ) V 2 = 0. 



(1.88) 
(1.89) 



Eqs. (1.88) and (1.89) can be rewritten as matrix Eq. (1.3) with 



[M] 



1 0 
0 1 



and [N] 



(Y l + Y 3 ) Y 3 

~gm + Y 3 -(72 + ^3) 



Since matrix [M] is nonsingular, such a two-port network can be described by a system of 
F-parameters as 



[Y] = -[M]~ l [N] = 



Yi + Y 3 -Y 3 
g m -Y 3 Y 2 + Y 3 



(1.90) 



Similarly, for a two-port network in the form of a 7-circuit using Kirchhoff 's voltage law or 
applying Eqs. (1.8) and (1.9), we obtain 



[Z] = - [My 1 [N] 



Zi + z 3 z 3 

I'm + Z 3 Z 2 + Z 3 



(1.91) 



If g m — 0 for a it -circuit and r m = 0 for a T-circuit, their corresponding matrices in a system of 
ABCZ>-parameters can be written as, for 7r-circuit, 



[ABCD] 



1 + 



Yi 



1 



Y\Y 2 Yi 
Y 1 + Y 2+ — 1 + - 



(1.92) 



PRACTICAL TWO-PORT NETWORKS 



23 




FIGURE 1.8 Equivalence of it- and T-circuits. 



for T-circuit, 



[ABCD] 



1+ 2 
1 

. Y 3 



Z!+Z 2 + 



Z x Zi 



1 + 



Zi 

z 3 



(1.93) 



For the appropriate relationships between impedances of a 7-circuit and admittances of a it -circuit, 
these two circuits become equivalent with respect to the effect on any other two-port network. For a 
it -circuit shown in Figure 1.8(a), we can write 



h = r 1 V a + Y 3 V n = Y 1 V 13 + Y 3 (Vu - V 23 ) = (Fi + Y 3 ) V l3 - Y 3 V 23 
h = Y 2 V 23 - Y 3 V l2 = Y 2 V 23 - Y 3 (Vu - V 23 ) = -Y 3 V l3 + (Y 2 + Y 3 ) V 23 . 

Solving Eqs. (1.94) and (1.95) for voltages V i 3 and V 23 yields 

Y 2 + Y 3 . . Y 3 



Vl3 

v 23 



Y\Y 2 + YJ 2 + YJ 2 

Y 3 



Y\Y 2 + F1I2 + Y\Y 2 

Yj + Y 3 
YxY 2 + YJ 2 + Y x Y 2 



YiY 2 + Y X Y 2 + Fi7 2 
Similarly, for a T-circuit shown in Figure 1.8(A), 

V a = Z1/1 + Z3/3 = Z1/1 + Z 3 (/j + 7 2 ) = (Z, + Z 3 )/i + Z 3 / 2 
Vb = Zj/j + Z3/3 = Z1/1 + Z 3 (/1 + / 2 ) = Z3/1 + (Z 2 + z 3 ) / 2 

and the equations for currents / t and I 2 can be obtained by 

Z 2 + Z 3 „ Z 3 



/2 = 



Z\Z 2 + Z\Z 2 + Z\Z 2 

z 3 

ZiZ 2 + Z X Z 2 + ZiZ 



v 13 + 



Z\Z 2 + Z\Z 2 + Z\Z 2 

z t + z 3 



Z\Z 2 + Z\Z 2 + Z\Z 2 



(1.94) 
(1.95) 



(1.96) 
(1.97) 



(1.98) 
(1.99) 



(1.100) 
(1.101) 



To establish a 7- to 7r -transformation, it is necessary to equate the coefficients for V\ 3 and V 23 
in Eqs. (1.100) and (1.101) to the corresponding coefficients in Eqs. (1.94) and (1.95). Similarly, to 
establish a n- to r-transformation, it is necessary to equate the coefficients for/i and/ 2 in Eqs. (1.98) 



24 PASSIVE ELEMENTS AND CIRCUIT THEORY 



TABLE 1.2 Relationships Between n- and T-Circuit Parameters. 



T 
/ - 


to 7T -Transl ortnation 


JT- 


to ^-Transformation 


Y\ 


z 2 


Zi = 


Y 2 


Z\Zi + Z2Z3 + Z\Z^ 


YiY 2 + Y 2 Y 3 + Yi¥ 3 


Y 2 


Z\ 


z 2 = 


Yi 


Z\ Zi + Z2Z3 + Z\ Z3 


YiY 2 + Y 2 Y 3 + YiY 3 


Y 3 


z 3 


z 3 = 


Yi 


Z1Z2 + Z2Z3 + Z1Z3 


YiY 2 + Y 2 Y 3 + Yi Y 3 



and (1.99) to the corresponding coefficients in Eqs. (1.96) and (1.97). The resulting relationships 
between admittances for a it -circuit and impedances for a 7-circuit are given in Table 1.2. 

1.5 THREE-PORT NETWORK WITH COMMON TERMINAL 

The concept of a two-port network with two independent sources can generally be extended to any 
multi-port networks. Figure 1.9 shows the three-port network where all three independent sources are 
connected to a common point. The three-port network matrix Eq. (1.3) can be described in a scalar 
form as 

muVi + fn l2 V 2 + mi 3 V 3 + nnh + "12/2 + "13/3 = 0 1 

»i 2 i Vi + m 22 V 2 + m 23 V 3 + n 2l Ii + « 2 2^2 + "23^3 = 0 > . (1.102) 

»2 31 Vi + m 3 2V2 + ffJ 33 V3 + n 3l I { + n 32 I 2 + n 33 I 3 — 0 J 

If matrix [N] in Eq. (1.102) is nonsingular when \N\ ^ 0, this system of three equations can 
be rewritten in admittance matrix representation in terms of the voltage matrix [V], similarly to a 
two-port network, by 



~h 




"I'll 


Y l2 


Y 13 




"V, 




h 




Y 21 


Y22 


Y 23 




v 2 


(1.103) 


h 




Y 31 


Y 32 


Y 33 




v 3 





The matrix [Y] in Eq. (1.103) is the indefinite admittance matrix of the three-port network and 
represents a singular matrix with two important properties: the sum of all terminal currents entering 
the circuit is equal to zero, that is, h + I 2 + I 3 — 0; and all terminal currents entering the circuit 



I 



LN 



-o- 

2 



0 

FIGURE 1.9 Basic diagram of three-port network with common terminal. 



THREE-PORT NETWORK WITH COMMON TERMINAL 



25 



depend on the voltages between circuit terminals, which makes the sum of all terminal voltages equal 
to zero, that is, V 13 + V 32 + V 2 i = 0. 

According to the first property, adding the left and right parts of matrix Eq. (1 . 103) results in 



On + Y 21 + Y 3l ) V l + (Y n + Y 22 + Y 32 ) V 2 + (Y 13 + Y 23 + Y 33 ) V 3 = 0. 



(1.104) 



Since all terminal voltages (V\ , V 2 , and V 3 ) can be set independently from each other, Eq. ( 1 . 104) 
can be satisfied only if any column sum is identically zero, 



r„ + F 21 + y 31 = o' 

Y n + Y 22 + Y 32 = 0 
Yi 3 + Y 23 + Y 33 = 0 



(1.105) 



The neither terminal currents will neither decrease nor increase, with the simultaneous change of 
all terminal voltages, by the same magnitude. Consequently, if all terminal voltages are equal to a 
nonzero value when V i — V 2 — V 3 — Vg, a lack of the terminal currents occurs when / t — I 2 — I 3 
— 0. For example, from the first row of the matrix Eq. (1.103) it follows that I\ — Y n Vi + Y \ 2 V 2 + 
F13V3; then we can write 



0 = (Y n + Y l2 + Y 13 )Vo 

which results due to the nonzero value Vo in 

Yn + Y l2 + Y u = 0. 

Applying the same approach to other two rows results in 

Y u + Y l2 + Y l3 = 0 ) 
Y21 + Y 22 + Y 23 = 0 . 
Y 31 + Y 32 + Y 33 = 0 



(1.106) 



(1.107) 



(1.108) 



Consequently, by using Eqs. (1.105) through (1.108), the indefinite admittance F-matrix of three- 
port network can be rewritten by 



[Y] 



Yn 
Y 2l 
(Yu + Y 2l ) 



Y l2 -(Yn + Yn) 

Y 22 — (Y 21 + F22) 

■ (Yn + Y 22 ) Y\\ + Y\ 2 + Y 2 \ + Y 22 



(1.109) 



By selecting successively terminal 1, 2, and 3 as the datum terminal, the corresponding three 
two-port admittance matrices of the initial three-port network can be obtained. In this case, the 
admittance matrices will correspond to a common emitter configuration shown in Figure 1.10(a), a 
common base configuration shown in Figure 1.10(A), and a common collector configuration of the 




different 



common terminals. 



26 PASSIVE ELEMENTS AND CIRCUIT THEORY 

TABLE 1.3 Y- and Z-Parameters of Active Device with Different Common Terminal. 



/"-Parameters Z-Parameters 



Common emitter 


I'll Y n 






Z11 


Z12 




(source) 


Y21 Y22 






Z21 






Common base (gate) 


Yn+Y n + Y2i + 


Y22 — 


(Y12 + Y22) 


Zn 


+ Z12 + Z21 + Z22 - 


-(Z12 + Z22) 




- (Y21 + Y22) 




Y22 




- (Z21 + Z22) 


Z22 


Common collector 


Yn 


-(Y U 


+ Y12) 




Z11 -(Zj 


1 + Z12) 


(drain) 


-(Y n + Y 2 i) Yi 


+ Y\2 - 


h Y21 + Y22 


- (Zn + Z21) Zn + Z12 


+ Z21 + Z22 



bipolar transistor shown in Figure 1.10(c), respectively. If the common emitter device is treated as a 
two-port network characterized by four F-parameters (I'll, Y12, F 2 i, and Y22), the two-port matrix of 
the common collector configuration with grounded collector terminal is simply obtained by deleting 
the second row and the second column in matrix Eq. (1.109). For the common base configuration 
with grounded base terminal, the first row and the first column should be deleted because the emitter 
terminal is considered the input terminal. 

A similar approach can be applied to the indefinite three-port impedance network. This allows the 
Z-parameters of the impedance matrices of the common base and the common collector configurations 
through known impedance Z-parameters of the common emitter configuration of the transistor to be 
determined. Parameters of the three-port network that can describe the electrical behavior of the 
three-port bipolar or field-effect transistor configured with different common terminals are given in 
Table 1.3. 



1.6 LUMPED ELEMENTS 

Generally, passive RF and microwave lumped or integrated circuits are designed based on the lumped 
elements, distributed elements, or combination of both types of elements. Distributed elements repre- 
sent any sections of the transmission lines of different lengths, types, and characteristic impedances. 
The basic lumped elements are inductors and capacitors that are small in size in comparison with the 
transmission-line wavelength A, and usually their linear dimensions are less than A/10 or even A/16. 
In applications where lumped elements are used, their basic advantages are small physical size and 
low production cost. However, their main drawbacks are lower quality factor and power-handling 
capability compared with distributed elements. 



1.6.1 Inductors 

Inductors are lumped elements that store energy in a magnetic field. Lumped inductors can be realized 
using several different configurations, such as a short-section of a strip conductor or wire, a single 
loop, or a spiral. The printed high-impedance microstrip-section inductor is usually used for low 
inductance values, typically less than 2 nH, and often meandered to reduce the component size. 
The printed microstrip single-loop inductors are not very popular due to their limited inductance per 
unit area. The approximate expression for the microstrip short-section inductance in free space is 
given by 



L (nH) = 0.2 x 10~ J / 



In 



/ \ W + t~ 

+ 1.193 + 



W + t 



3/ 



(1.110) 



LUMPED ELEMENTS 27 




1^ 





(</) 



FIGURE 1.11 Spiral inductor layouts. 



where the conductor length /, conductor width W, and conductor thickness / are in microns, and the 
term K g accounts for the presence of a ground plane, defined as 



K„ = 0.57 - 0.145 In - 



W 



W 

for — > 0.05 



(1.111) 



where h is the spacing from ground plane [6,7]. 

Spiral inductors can have a circular configuration, a rectangular (square) configuration shown in 
Figure 1.11(a), or an octagonal configuration shown in Figure 1.11 (ft), if the technology allows 45° 
routing. The circular geometry is superior in electrical performance, whereas the rectangular shapes 
are easy to lay out and fabricate. Printed inductors are based on using thin-film or thick-film Si or 
GaAs fabrication processes, and the inner conductor is pulled out to connect with other circuitry 
through a bondwire, an air bridge, or by using multilevel crossover metal. The general expression for 
spiral inductor, which is also valid for its planar integration within accuracy of around 3%, is based 
on Wheeler formula and can be obtained as 



L (nH) = 



l + K 2 p 



(1.112) 



where n is the number of turns, d mg = (d C ut + d\ a )l2 is the average diameter, p = (d 0M + di n )/(d 0M - 
d m ) is the fill ratio, d out is the outer diameter in [j.m, d m is the inner diameter in \im, and the coefficients 
Ki and K 2 are layout-dependent as follows: square — Ki — 2.34, K 2 — 2.75; hexagonal — K\ — 2.33, 
K 2 = 3.82; octagonal— K x = 2.25, K 2 = 3.55 [8,9]. 

In contrast to the capacitors, high-quality inductors cannot be readily available in a standard 
complementary metal-oxide-semiconductor (CMOS) technology. Therefore, it is necessary to use 
special techniques to improve the inductor electrical performance. By using a standard CMOS tech- 
nology with only two metal layers and a heavily doped substrate, the spiral inductor will have a 
large series resistance, compared with three-four metal layer technologies, and the substrate losses 
become a very important factor due to a relatively low resistivity of silicon. A major source of 
substrate losses is the capacitive coupling when current is flowing not only through the metal strip, 
but also through the silicon substrate. Another important source of substrate losses is the inductive 
coupling when, due to the planar inductor structure, the magnetic field penetrates deeply into the 
silicon substrate, inducing current loops and related losses. However, the latter effects are partic- 
ularly important for large-area inductors and can be overcome by using silicon micromachining 
techniques [10]. 



28 



PASSIVE ELEMENTS AND CIRCUIT THEORY 



AAA 



o 



o 



A', 





FIGURE 1.12 Equivalent circuit of a square spiral inductor. 



The simplified equivalent circuit for the CMOS spiral microstrip inductor is shown in Figure 1.12, 
where L s models the self and mutual inductances, R s is the series coil resistance, C ox is the parasitic 
oxide capacitance from the metal layer to the substrate, i? s ; is the resistance of the conductive silicon 
substrate, C s ; is the silicon substrate parasitic capacitance, and C c is the parasitic coupling capacitance 
[11]. The parasitic silicon substrate capacitance C S j is sufficiently small, and in most cases it can be 
neglected. Such a model shows an accurate agreement between simulated and measured data within 
10% across a variety of inductor geometries and substrate dopings up to 20 GHz [12]. At frequencies 
well below the inductor self-resonant frequency cosrf, the coupling capacitance C c between metal 
segments due to fringing fields in both the dielectric and air regions can also be neglected since the 
relative dielectric constant of the oxide is small enough [13]. In this case, if one side of the inductor 
is grounded, the self-resonant frequency of the spiral inductor can approximately be calculated from 



At frequencies higher than self-resonant frequency o)skf, the inductor exhibits a capacitive behav- 
ior. The self-resonant frequency cosrf is limited mainly by the parasitic oxide capacitance C ox , which 
is inversely proportional to the oxide thickness between the metal layer and substrate. The frequency 
at which the inductor quality factor Q is maximal can be obtained as 



The inductor metal conductor series resistance R s can be easily calculated at low frequencies 
as the product of the sheet resistance and the number of squares of the metal trace. However, at 
high frequencies, the skin effect and other magnetic field effects will cause a nonuniform current 
distribution in the inductor profile. In this case, a simple increase in the diameter of the inductor metal 
turn does not necessarily reduce correspondingly the inductor series resistance. For example, for the 
same inductance value, the difference in resistance between the two inductors, when one of which 
has a two times wider metal strip, is only a factor of 1.35 [14]. Moreover, at very high frequencies, 
the largest contribution to the series resistance does not come from the longer outer turns, but from 
the inner turns. This phenomenon is a result of the generation of circular eddy currents in the inner 
conductors, whose direction is such that they oppose the original change in magnetic field. On the 




(1.113) 




(1.114) 



LUMPED ELEMENTS 29 



inner side of the inner turn, coil current and eddy current flow in the same direction, so the current 
density is larger than average. On the outer side, both currents cancel, and the current density is 
smaller than average. As a result, the current in the inner turn is pushed to the inside of the conductor. 

In hybrid or monolithic applications, bondwires are used to interconnect different components 
such as lumped elements, planar transmission lines, solid-state devices, and integrated circuits. These 
bondwires, which are usually made of gold or aluminium, have 0.5- to 1.0-mil diameters, and their 
lengths are electrically shorter compared with the operating wavelength. To characterize the electrical 
behavior of the bondwires, simple formulas in terms of their inductances and series resistances 
can be used. As a first-order approximation, the parasitic capacitance associated with bondwires 
can be neglected. When / 3> d, where / is the bondwire length in p.m and d is the bondwire diameter 
in [im, 



where C — 0.25 tanh (48/d) is the frequency-dependent correction factor, which is a function of 
bondwire diameter and its material's skin depth S [8,15]. 

1.6.2 Capacitors 

Capacitors are lumped elements that store energy due to an electric field between two electrodes (or 
plates) when a voltage is applied across them. In this case, charge of equal magnitude but opposite 
sign accumulates on the opposing capacitor plates. The capacitance depends on the area of the plates, 
separation, and dielectric material between them. The basic structure of a chip capacitor shown in 
Figure 1.13(a) consists of two parallel plates, each of area A — W x / and separated by a dielectric 
material of thickness d and permittivity e 0 e r , where £o is the free-space permittivity (8.85xl0~ 12 
farads/m) and e T is the relative dielectric constant. 




(1.115) 



/ 





♦ 




tb) 

FIGURE 1.13 Parallel capacitor topology and its equivalent circuit. 



30 



PASSIVE ELEMENTS AND CIRCUIT THEORY 



Chip capacitors are usually used in hybrid integrated circuits when relatively high capacitance 
values are required. In the parallel-plate configuration, the capacitance is commonly expressed as 

Wl 

C (pF) = 8.85 x lfr 3 e r — (1.116) 
d 

where W, I, and d are dimensions in millimeters. Generally, the low-frequency bypass capacitor 
values are expressed in microfarads and nanofarads, high-frequency blocking and tuning capacitors 
are expressed in picofarads, and parasitic or fringing capacitances are written in femtofarads. This 
basic formula given by Eq. (1.116) can also be applied to capacitors based on a multilayer technique 
[7]. The lumped-element equivalent circuit of a capacitor is shown in Figure 1.13(A), where L s is the 
series plate inductance, R s is the series contact and plate resistance, and C p is the parasitic parallel 
capacitance. When C>C p , the frequency a) ssa s, at which the reactances of series elements C and L s 
become equal, is called the capacitor self-resonant frequency, and the capacitor impedance is equal 
to the resistance R s . 

For monolithic applications, where relatively low capacitances (typically less than 0.5 pF) are 
required, planar series capacitances in the form of microstrip or interdigital configurations can be 
used. These capacitors are simply formed by gaps in the center conductor of the microstrip lines, 
and they do not require any dielectric films. The gap capacitor, shown in Figure 1.14(a), can be 
equivalently represented by a series coupling capacitance and two parallel fringing capacitances [16]. 
The interdigital capacitor is a multifinger periodic structure, as shown in Figure 1.14(A), where the 
capacitance occurs across a narrow gap between thin-film transmission-line conductors [17], These 
gaps are essentially very long and folded to use a small amount of area. In this case, it is important to 
keep the size of the capacitor very small relative to the wavelength, so that it can be treated as a lumped 
element. A larger total width-to-length ratio results in the desired higher shunt capacitance and lower 
series inductance. An approximate expression for the total capacitance of interdigital structure, with 
s — W and length / less than a quarter wavelength, can be given by 

C (pF) = (e r + 1)1 [(N - 3) A { + A 2 ] (1.117) 




(O 



FIGURE 1.14 Different series capacitor topologies. 



TRANSMISSION LINE 31 



where N is the number of fingers and 

Ai (pF/nm) = 4.409 tanh 

A 2 (pF/nm) = 9.92 tanh 



0.55 



0.52 



W 



w 



x 10" 



x 10" 



(1.118) 



(1.119) 



where h is the spacing from the ground plane. 

Series planar capacitors with larger values, which are called the metal-insulator-metal (MIM) 
capacitors, can be realized by using an additional thin dielectric layer (typically less than 0.5 \im) 
between two metal plates, as shown in Figure 1.14(c) [7]. The bottom plate of the capacitor uses 
a thin unplated metal, and typically the dielectric material is silicon nitride (Si3N 4 ) for integrated 
circuits on GaAs and SiC>2 for integrated circuits on Si. The top plate uses a thick-plated conductor 
to reduce the loss in the capacitor. These capacitors are used to achieve higher capacitance values in 
small areas (10 pF and greater), with typical tolerances from 10% to 15%. The capacitance can be 
calculated according to Eq. (1.116). 



1.7 TRANSMISSION LINE 

Transmission lines are widely used in matching circuits in power amplifiers, in resonant and feedback 
circuits in the oscillators, filters, directional couplers, power combiners, and dividers. When the 
propagated signal wavelength is compared to its physical dimension, the transmission line can be 
considered as a two-port network with distributed parameters, where the voltages and currents vary 
in magnitude and phase over length. 

Schematically, a transmission line is often represented as a two-wire line, as shown in Figure 
1.15(a), where its electrical parameters are distributed along its length. The physical properties of a 
transmission line are determined by four basic parameters: 

1. The series inductance L due to the self-inductive phenomena of two conductors. 

2. The shunt capacitance C in view of the close proximity between two conductors. 

3. The series resistance R due to the finite conductivity of the conductors. 

4. The shunt conductance G that is related to the dielectric losses in the material. 

As a result, a transmission line of length Ax represents a lumped-element circuit shown in 
Figure 1.15(Z>), where AL, AC, AR, and AG are the series inductance, the shunt capacitance, the 
series resistance, and the shunt conductance per unit length, respectively. If all these elements are 
distributed uniformly along the transmission line, and their values do not depend on the chosen 
position of Ax, this transmission line is called the uniform transmission line. Any finite length of the 
uniform transmission line can be viewed as a cascade of section length Ax. 

To define the distribution of the voltages and currents along the uniform transmission line, it is 
necessary to write the differential equations using Kirchhoff 's voltage law for instantaneous values 
of the voltages and currents in the line section of length A.v, distant x from its beginning. For the 
sinusoidal steady-state condition, the telegrapher equations for V(x) and I(x) are given by 

y 2 V(x) = 0 (1.120) 
y 2 I(x) = 0 (1.121) 



d 2 V(x) 
dx 2 
d 2 I(x) 
dx 2 



32 



PASSIVE ELEMENTS AND CIRCUIT THEORY 



/ - \l 




r+ AC 



(«) 



V \v 



/+ Al 




x + A.v 



FIGURE 1.15 Transmission line schematics. 



where y — a + jfi — ^/(AR + jcoAL)(AG + jcoAC) is the complex propagation constant (which 
is a function of frequency), a is the attenuation constant, and fi is the phase constant. The general 
solutions of Eqs. (1.120) and (1.121) for voltage and current of the traveling wave in the transmission 
line can be written as 



V(x) — A i exp(— yx) + A 2 e.xp(yx) 
I(x) = ^-exp(-yx) - ^exp(yx) 



(1.122) 
(1.123) 



where Z 0 = *J(AR + jcoAL) / (AG + jcoAC) is the characteristic impedance of the transmission 
line, Vi — Aiexp(— yx) and V r — A2exp(yx) represent the incident voltage and the reflected voltage, 
respectively, and / ; = A [ exp(— yx)/Z 0 and / r = A 2 exp(yx)/Z 0 are the incident current and the reflected 
current, respectively. From Eqs. (1.122) and (1.123) it follows that the characteristic impedance of the 
transmission line Z 0 represents the ratio of the incident (reflected) voltage to the incident (reflected) 
current at any position on the line as 



Vi(x) = VAx) 

h(x) I T (X) ' 



(1.124) 



For a lossless transmission line, when R = G = 0 and the voltage and current do not change 
with position, the attenuation constant a — 0, the propagation constant y —jf} — ja> V AL AC, and 
the phase constant /3 — coV AL AC. Consequently, the characteristic impedance is reduced to Z 0 = 
■JL I C and represents a real number. The wavelength is defined as X — lit/ft — 2jt/(d^/ AL AC and 
the phase velocity as v p — co//3 — 1 / V AL AC. 

Figure 1.16 represents a transmission line of characteristic impedance Z 0 terminated with a load 
Z L . In this case, the constants Ai and A 2 are determined at the position x — I by 

V(l) = A l exp(-yl) + A 2 exp(yl) (1.125) 

/(/) = ^exp (-//)- ^exp(y/) (1.126) 



TRANSMISSION LINE 33 



and equal to 



1(0) 



1(1) 



► 




► 


t 

'(0) 


k 

r 




i 


v. 

»'(/) 









.t-0 x - I 

FIGURE 1.16 Loaded transmission line. 



V(l) + ZqIQ) 
Ai = exp(yZ) 

V(Z)-Z o /0) 
A 2 = exp(-yZ). 



(1.127) 
(1.128) 



As a result, wave equations for voltage V(x) and current I(x) can be rewritten as 



V(l)+Z 0 I(l) V(l) - Z 0 /(/) 
V(.\-) = exp [y (I - x)] + exp [-y (I - x)] 

V(l)+Z 0 I(l) V(l) - Z 0 /(/) 
/ (x) = — exp [y (I - x)] — exp [-/ (/ - x)] 



2Z„ 



2Z„ 



(1.129) 
(1.130) 



which allows their determination at any position on the transmission line. 

The voltage and current amplitudes at x — 0 as functions of the voltage and current amplitudes at 
x — I can be determined from Eqs. (1.129) and (1.130) as 



VQ) + Z 0 /(0 - Z 0 I(I) 
V(0)= exp(y/)+ exp(-yl) (1.131) 

no + z 0 /(/) v(/)-z 0 /(/) 

7(0)= — exp(y/) — exp(— yl) . (1.132) 



2Z n 



2Z 0 



By using the ratios cosh x — [exp(.v) + exp(— x)]/2 and sinh x — [exp(x) — exp(— x)]/2, Eqs. 
(1.131) and (1.132) can be rewritten in the form 



V(0) = V / (/)cosh(j//) + Z 0 7(/)sinh(y/) 



K0) 



sinh(j//) + 7(/)cosh(y/) 



(1.133) 
(1.134) 



which represents the transmission equations of the symmetrical reciprocal two-port network expressed 
through the ABC7)-parameters when AD -BC — 1 andA — D. Consequently, the transmission ABCD- 
matrix of the lossless transmission line with a — 0 can be given by 



[ABCD] = 



cos 6* jZ 0 sin ( 
j sin 8 



(1.135) 



34 PASSIVE ELEMENTS AND CIRCUIT THEORY 



Using the formulas to transform ABCZ>-parameters into S-parameters yields 



[S] = 



o exp(-ye) 

exp(-jO) 0 



(1.136) 



where 6 — fil is the electrical length of the transmission line. 

In the case of the loaded lossless transmission line, the reflection coefficient T is defined as the 
ratio between the reflected voltage wave and the incident voltage wave, given at x as 

rW=§ = 4 i exp(2i^). (1.137) 

By taking into account Eqs. (1.127) and (1.128), the reflection coefficient for x — I can be defined 

as 

Z - Z 0 

r = (1.138) 

Z + Zq 

where T represents the load reflection coefficient and Z = Zl = V(t)/I(l). If the load is mismatched, 
only part of the available power from the source is delivered to the load. This power loss is called the 
return loss (RL), and is calculated in decibels as 

RL = -201og 10 |r| . (1.139) 

For a matched load when r = 0, a return loss is of oo dB. A total reflection with r = 1 means a 
return loss of 0 dB when all incident power is reflected. 

According to the general solution for voltage at a position x in the transmission line, 

V(x) = Vi(x) + V r (x) = Vdl + T(x)] . (1.140) 

Hence, the maximum amplitude (when the incident and reflected waves are in phase) is 

Vnmto = IVil [1 + |r(*)|] (1.141) 

and the minimum amplitude (when these two waves are out of phase) is 

V^(x)=\Vi\H-\r(x)\]. (1.142) 

The ratio of V max to V^, which is a function of the reflection coefficient T, represents the voltage 
standing wave ratio (VSWR). The VSWR is a measure of mismatch and can be written as 

VSWR = = — (1.143) 

v mm i — in 

which can change from 1 to oo (where VSWR — 1 implies a matched load). For a load impedance 
with zero imaginary part when Zl = Rl, the VSWR can be calculated using VSWR — Rl/Zq when 
R L > Z 0 and VSWR = Z 0 /R L when Z 0 > R L . 

From Eqs. (1.133) and (1.1 34) it follows that the input impedance of the loaded lossless transmis- 
sion line can be obtained as 

m z^+jz^m 

1(0) Z 0 + jZ h tan (0) 



which gives an important dependence between the input impedance, the transmission-line parameters 
(electrical length and characteristic impedance), and the arbitrary load impedance. 



TYPES OF TRANSMISSION LINES 



35 



1.8 TYPES OF TRANSMISSION LINES 

Several types of transmission lines are available when designing RF and microwave active and passive 
circuits. Coaxial lines have very high bandwidth and high power-handling capability, and are widely 
used for impedance transformers and power combiners. Planar transmission lines as an evolution 
of the coaxial and parallel-wire lines are compact and readily adaptable to hybrid and monolithic 
integrated circuit fabrication technologies at RF and microwave frequencies [18]. If coaxial line is 
deformed in such a manner that both the center and outer conductors are square or rectangular in 
cross-section, and then if side walls of the rectangular coaxial system are extended to infinity, the 
resultant flat-strip transmission system would have a form factor that is adaptable to the printed-circuit 
technique. Similarly, if the parallel-wire line is replaced by its equivalent of a single wire and its image 
in a conducting ground plane, and if this single wire is, in turn, progressively distorted into a flat strip, 
the resulting transmission system is again a planar structure. There is an important aspect that differ 
flat-strip transmission lines from coaxial lines. In a coaxial line, an impedance discontinuity acts as a 
shunt capacitance, while a discontinuity in a flat strip has a series inductance in its equivalent circuit. 
Holes and gaps in center conductor strips also represent discontinuities that can be utilized in many 
applications to microwave circuitry. 

1.8.1 Coaxial Line 

A main type of wave propagated along a coaxial line shown in Figure 1.17 is the transverse electro- 
magnetic (TEM) wave. When the transverse fields of a TEM wave are the same as the static fields that 
can exist between the conductors, the electromagnetic properties of a coaxial line can be characterized 
by the following parameters [19]: the shunt capacitance per unit length 



where a is the radius of inner conductor and b is the inner radius of outer conductor; the series 
inductance per unit length 




(1.145) 




(1.146) 




FIGURE 1.17 Coaxial line structure. 



36 PASSIVE ELEMENTS AND CIRCUIT THEORY 



where fi 0 — 4ttx 10 7 H/m is the permeability of free space and /x r is the relative magnetic constant 
or substrate permeability; the series resistance per unit length 

*=to{b + a) (U47) 

where R s — pi A(f ) — -Jn^Qpf is the surface resistivity, p is the metallization electrical resistivity, 
A(f) is the penetration depth, and/ is the frequency; the shunt conductance per unit length 

G = 2jr<x/ln (-^ = 2na>e 0 e t tan5/ln (-J (1.148) 

where a is the dielectric conductivity and tan<5 is the dielectric loss tangent; the characteristic 
impedance 

Z 0 = ^-lnf-} (1.149) 

where r) — -/Ji/b is the wave impedance of the lossless coaxial line identical to the intrinsic impedance 
of the medium. 

The conductor loss factor (in Np/m) can be written as 

a c =^=- (1.150) 

whereas the dielectric loss factor (in Np/m) can be written as 

GZ 0 ar\ tan<5 

Q! d =—— = — = Tlt/Sr—- (1.151) 
L L a 

where X is the free-space wavelength. 



1.8.2 Stripline 

The geometry of a commonly used stripline is shown in Figure 1.18. The strip conductor of width W 
is placed between two flat dielectric substrates with the same dielectric constant. The outer surfaces 
of these substrates are metallized and serve as a ground conductor. In practice, the strip conductor 
is etched on one of the dielectric substrates by photolithography process. Since the stripline has two 
conductors and a homogeneous dielectric, it can support a pure TEM propagation mode, which is the 
usual mode of operation. The advantages of striplines are good electromagnetic shielding and low 
attenuation losses, which make them suitable for high-g and low-interference applications. However, 
striplines require strong symmetry that makes their tuning complicated due to difficult access to 
center conductor. As a result, the stripline structure is not convenient for incorporating chip elements 
and associated bias circuitry. 

The exact expression for the characteristic impedance of a lossless stripline of zero thickness is 
given by 

30;r K (k) 

Z 0 = — (1.152) 

JTrK(k>) 



TYPES OF TRANSMISSION LINES 37 




















W 
< *■ 






FIGURE 1.18 Stripline structure. 



where 



7T/2 



dtp 



y/l — k 2 sin 2 <p 



(1.153) 



is the complete elliptic integral of the first kind, k — sech (nW/2b), and k' — -n/1 — k 2 [20,21]. 
An expression for the ratio K(k)/K(k') can be simplified to 



K(k) 
K(k>) 



l + l 

7r/ln 2 ^— forO <k< — 

1 - VP / " " V2 



iln( 2 i±^ 

71 \ 1 - V* 



(1.154) 



for — — < A < 1 
V2 



which provides the relative error lower than 3 x 10~ 6 [22]. 

In practice, it makes sense to use a sufficiently simple formula without complicated special 
functions [23]. In this case, the formula for Z 0 can be written within 1% of the exact results as 



Z 0 = 



30jt 



fs t W e +0.4416 
where W e is the effective width of the center conductor defined by 



(1.155) 



W e _ W 

~b ~ ~b 



0 



0.35 



W 



w 

for — > 0.356 
b 

W 

for — < 0.356 
b 



(1.156) 



For a stripline with a TEM propagation mode, the dielectric loss factor is the same as for 
coaxial line, which is determined by Eq. (1.151). An approximation result for the conductor loss 
factor a c (in Np/m) can be obtained by 



2.7 x 10- 3 fl s e r Z 0 ^_ 
A — — ^— - forZoV^<120 



30tt (b - t) 
0.16# s 



(1.157) 



B 



Znbn 



for Z 0 ^/e^ > 120 



38 PASSIVE ELEMENTS AND CIRCUIT THEORY 



100 



0.1 





Er-2 




























^4 


















































































— - 






























1 0 




























20 





















































































































V 




























































s 































































W/b 



10 



FIGURE 1.19 Stripline characteristic impedance versus W/b. 



with 



2W 1 b + t 2b -t 
1 + + ln- 



B = 1 + 



b - t it b - t 
b 



0.5W + 0.7? 



0.414? 1 AnW\ 



(1.158) 
(1.159) 



where t is the thickness of the strip [4], 

Figure 1.19 shows the characteristic impedance Zq of a stripline as a function of the normalized 
strip width W/b for various e r according to Eqs. (1.155) and (1.156). Typical values of the main 
electrical and thermal properties of some substrate materials are listed in Table 1.4. 



TABLE 1.4 Electrical and Thermal Properties of Substrate Materials. 




Dielectric Constant, 


Loss Tangent, 


Coefficient of Thermal 


Typical Substrate 


£ r @ 10 GHz 


tanS @ 10 GHz 


Expansion (ppm/°C) 


Alumina 99.5% 


9.8 


0.0003 


6.7 


Aluminum nitride 


8.7 


0.001 


4.5 


Barium tetratitanade 


37 


0.0002 


8.3 


Beryllia 99.5% 


6.6 


0.0003 


7.5 


Epoxy glass FR-4 


4.7 


0.01 


3.0 


Fused quartz 


3.78 


0.0001 


0.5 


Gallium arsenide 


13.1 


0.0006 


6.5 


Silicon 


11.7 


0.004 


4.2 


Teflon 


2.5 


0.0008 


15 



TYPES OF TRANSMISSION LINES 39 



1.8.3 Microstrip Line 

In a microstrip line, the grounded metallization surface covers only one side of dielectric substrate, 
as shown in Figure 1 .20. Such a configuration is equivalent to a pair-wire system for the image of the 
conductor in the ground plane which produces the required symmetry [24]. In this case, the electric 
and magnetic field lines are located in both the dielectric region between the strip conductor and 
the ground plane and in the air region above the substrate. As a result, the electromagnetic wave 
propagated along a microstrip line is not a pure TEM, since the phase velocities in these two regions 
are not the same. However, in a quasistatic approximation, which gives sufficiently accurate results as 
long as the height of the dielectric substrate is very small compared with the wavelength, it is possible 
to obtain the explicit analytical expressions for the electrical characteristics. Since microstrip line is 
an open structure, it has a major fabrication advantage over the stripline due to simplicity of practical 
realization, interconnection, and adjustments. 

The exact expression for the characteristic impedance of a lossless microstrip line with finite strip 
thickness is given by [25,26] 



60 / 8/i W e 
— — In 1 

\W e 4/i 



i20;r \w t /w e 

— — + 1.393 + 0.667 In — + 1.444 
V^re \_ h \ h 



c w , 

for — < 1 
h ~ 

<■ W , 
for — > 1 

h ~ 



(1.160) 



where 



W e _ W AW 
~h~lt + ~h~ 



AW 



1.25 t ( 4nW\ W 

1 + In — ^ — J fory<l/2jr 



Ti h 



1.25 t ( 2h 

1 + ln— 

ti h V t 



W 

for — > 1/271- 
h 



(1.161) 



(1.162) 



e r + 1 s r — 1 



1 e t -l t h 

2~ VI + UY/W 4^6~hi W' 



(1.163) 



Figure 1.21 shows the characteristic impedance Zo of a microstrip line with zero strip thickness 
as a function of the normalized strip width Wlh for various e r according to Eqs. (1.160) to (1.163). 




FIGURE 1.20 Microstrip line structure. 



40 



PASSIVE ELEMENTS AND CIRCUIT THEORY 




0.1 I lti 

FIGURE 1.21 Microstrip characteristic impedance versus W/h. 



W'h 



In practice, it is possible to use a sufficiently simple formula to estimate the characteristic 
impedance Zq of a microstrip line with zero strip thickness written as [27] 



Zn = 



120tt h 



1 



fs t W 1 + 1.735s- 0 - 0724 (M7 /i)- 



(1.164) 



For a microstrip line in a quasi-TEM approximation, the conductor loss factor k c (in Np/m) as a 
function of the microstrip-line geometry can be obtained by 



1.38A 



R s 32 -{WJhf 
hZ 0 32 + (W e /h) 2 



, R s Z 0 s K [W t 0.667 WJ h 
6.1 ■ 10" 5 A^-!- ' ' 1 



1.444 + W e /h 



W 

for — < 1 

h ~ 

W 

for — > 1 

h ~ 



(1.165) 



with 



h ( 1 2B\ 
1 + — 1 + -In — 

W e \ Tt t ) 

w 

2nW for— <\/2ir 
h 

W 

h for — > 1/2jt 
h 



(1.166) 



(1.167) 



where WJh is given by Eqs. (1.161) and (1.162) [28]. 

The dielectric loss factor (in Np/m) can be calculated by 



cii — 27.3 



1 tanS 



£ r — 1 -/£ re A 



(1.168) 



Conductor loss is a result of several factors related to the metallic material composing the ground 
plane and walls, among which are conductivity, skin effect, and surface ruggedness. For most mi- 
crostrip lines (except some kinds of semiconductor substrate such as silicon), the conductor loss is 



TYPES OF TRANSMISSION LINES 41 



lABLE 1.5 


Electrical Resistivity of Conductor Materials. 










Elcctricsl Resistivity 






Electrical Resistivity 


Material 


Symbol 


( |_i j~2 cm) 


Material 


Symbol 


( \a£2 cm) 


Aluminum 


Al 


2.65 


Palladium 


Pd 


10.69 


Copper 


Cu 


1.67 


Platinum 


Pt 

r I 


1 n fo 


Gold 


Au 


2.44 


Silver 


Ag 


1.59 


Indium 


In 


15.52 


Tantalum 


Ta 


15.52 


Iron 


Fe 


9.66 


Tin 


Sn 


11.55 


Lead 


Pb 


21.0 


Titanium 


Ti 


55.0 


Molybdenum 


Mo 


5.69 


Tungsten 


W 


5.6 


Nickel 


Ni 


8.71 


Zinc 


Zn 


5.68 



much more significant than the dielectric loss. The conductor losses increase with increasing char- 
acteristic impedance due to greater resistance of narrow strips. The electrical resistivity of some 
conductor materials is given in Table 1.5. 



1.8.4 Slotline 

Slotlines are usually used when it is necessary to realize a high value of the characteristic impedance 
Z 0 [29,30]. A slotline is dual to a microstrip line and represents a narrow slot between two conductive 
surfaces, one of which is grounded. Changing the width of the slot can easily change the characteristic 
impedance of the slotline. The transverse electric //-mode wave propagates along the slotline. Three 
basic types of slotlines are unilateral, antipodal, and bilateral. The geometry of a unilateral slotline 
is shown in Figure 1.22, with a narrow gap in the conductive coating on one side of the dielectric 
substrate and being bare on the other side of substrate. Slotline can be used either alone or with 
microstrip line on the opposite side of substrate. 

It is difficult to provide exact analytical expressions to calculate the slotline parameters. However, 
an equation for Zq can be obtained for a quasi-TEM approximation with zero conductor thickness 
and infinite width of the entire slotline system as 



for 0.02 < Wlh < 0.2 




FIGURE 1.22 Slotline structure. 



42 



PASSIVE ELEMENTS AND CIRCUIT THEORY 




W/h 



0.02 0.1 1 

FIGURE 1.23 Slotline characteristic impedance versus W/h. 



for 0.2 < W/h < 1.0 



W Ji 
10.25- 2.171 lne r + —(2.1 - 0.617 lne r )- 10 2 - 
h X 



w ( w\ (w 

Z Q = 113.19 - 23. 257 lne r + 1.25— (114.59 - 22.531 lns r ) + 20 M - - I — -0.2 

0.15 + 0.1 lne r + ^ (0.899 lne r - 0.79) 
h 



(1.170) 



where 0.01 < h/X S 0.25/Vsr — 1 and A, is the free-space wavelength [31]. 

Figure 1 .23 shows the characteristic impedance Zq of a slotline within the error of 2% as a function 
of the normalized slot width W/h for h/X — 0.02 and various e r = 9.7, 1 1, 12, . . . , 20 calculated by 
Eqs. (1.169) and (1.170). 



1.8.5 Coplanar Waveguide 

A coplanar waveguide (CPW) is similar in structure to a slotline, the only difference being a third 
conductor centered in the slot region. The center strip conductor and two outer grounded conductors 
lie in the same plane on substrate surface, as shown in Figure 1.24 [32,33]. A coplanar configuration 
has some advantages such as low dispersion, ease of attaching shunt and series circuit components, no 
need for via holes, or simple realization of short-circuited ends, which makes a coplanar waveguide 
suitable for hybrid and monolithic integrated circuits. In contrast to the microstrip and stripline, the 
coplanar waveguide has shielding between adjacent lines that creates a better isolation between them. 
However, like microstrip and stripline, the coplanar waveguide can be also described by a quasi-TEM 



TYPES OF TRANSMISSION LINES 43 




FIGURE 1.24 Coplanar waveguide structure. 

approximation for both numerical and analytical calculations. Because of the high dielectric constant 
of the substrate, most of the RP energy is stored in the dielectric and the loading effect of the grounded 
cover is negligible if it is more than two slot widths away from the surface. Similarly, the thickness 
of the dielectric substrate with higher relative dielectric constants is not so critical, and practically it 
should be one or two times the width W of the slots. 

The approximate expression of the characteristic impedance Zq for zero metal thickness which is 
satisfactory accurate in a wide range of substrate thicknesses can be written as 



Zn = 



30tt K (£') 
"*■(*) 



(1.171) 



where 



1 + 



e r - 1 K(k') K(h) 
2 K(k) K (k[) 



(1.172) 



k\, and K is 



k = s/(s + 2W), h = (sinh jr.s/4/i)/(sinh it (s + 2W)/4h), k' = Vl - k 1 , k[ = y/i 
the complete elliptic integral of the first kind [34] . The values of ratios K(k)/K(k') and K(ki )/K(k[ ) can 
be defined from Eq. ( 1 . 1 54). Figure 1 .25 shows the characteristic impedance Z 0 of a coplanar waveg- 
uide as a function of the parameter s/(s + 2W) for various s t according to Eqs. (1.171) and (1.172). 



Ah n 



100 



10 



"""""-\_c,-l 

























s/{s + 2W) 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

FIGURE 1.25 Coplanar waveguide characteristic impedance versus s/(s + 2W). 



44 



PASSIVE ELEMENTS AND CIRCUIT THEORY 



1.9 NOISE 

The electrical performance of RF and microwave transmitters of different applications can be affected 
by many factors, with the effect of noise as one of the most fundamental. In this case, it is necessary 
to keep the ratio of average (or peak) signal power to average noise power so large that the noise in 
a transmitter path has no harmful effects on overall system performance. Several basic approaches 
can provide this where it is possible, such as powerful transmitters and high-gain antennas to develop 
large signals at the receiver input, stabilized oscillators with minimum phase noise, power amplifier 
and mixer circuits so that they introduce a minimum amount of additional noise when processing 
signals, and modulation and coding schemes that facilitate the separation of signal and noise. 

1.9.1 Noise Sources 

There are several primary noise sources in the electrical circuit. Thermal or white noise is created by 
random motion of charge carriers due to thermal excitation, being always found in any conducting 
medium whose temperature is above absolute zero whatever the nature of the conduction process or 
the nature of the mobile charge carriers [35]. This random motion of carriers creates a fluctuating 
voltage on the terminals of each resistive element which increases with temperature. However, if 
the average value of such a voltage is zero, then the noise power on its terminal is not zero being 
proportional to the resistance of the conductor and to its absolute temperature. The resistor as a 
thermal noise source can be represented by either of the noise sources shown in Figure 1.26. The 
noise voltage source and noise current source can be respectively described by Nyquist equations 
through their mean-square noise voltage and noise current values as 



where k — 1.38 x 10~ 23 J/K is the Boltzmann constant, T is the absolute temperature, and kT — 4 x 
10~ 21 W/Hz = — 174 dBm/Hz at ambient temperature T — 290 K. The thermal noise is proportional 
to the frequency bandwidth A/', and it can be represented by the voltage source in series with resistor 
R, or by the current source in parallel to the resistor R. The maximum noise power can be delivered to 
the load when R — R L , where R L is the load resistance, being equal to kTAf. Hence, the noise power 
density when the noise power is normalized by A/ is independent of frequency and is considered as 
white noise. The root-mean-square noise voltage and current are proportional to the square root of 
the frequency bandwidth A/. 

Shot noise is associated with the carrier injection through the device p-n junction, being generated 
by the movement of individual electrons within the current flow. In each forward biased junction, 



•1 



4kTRAf 
AkT Af 



(1.173) 



(1.174) 



R 



R 




o 



(a) 



(A) 



FIGURE 1.26 Equivalent circuits to represent thermal noise sources. 



NOISE 



45 



there is a potential barrier that can be overcome by the carriers with higher thermal energy. Such a 
process is random and mean-square noise current can be given by 



where q is the electron charge and / is the direct current flowing through the p-n junction. The shot 
noise depends on the thermal energy of the carriers near the potential barrier and its power density 
is independent of frequency. It has essentially a flat spectral distribution and can be treated as the 
thermal or white type of noise with current source i% connected in parallel to the small-signal junction 
resistance. In a voltage noise representation, when the noise voltage source is connected in series 
with such a resistor, it can be written as 



where = kT/ql is the junction resistance. 

Circuits containing more than one resistor can be analyzed by reducing their number to the only 
one (Thevenin) equivalent resistance to obtain the mean-square noise voltage in the form of Eq. 
(1.173) [36]. As an example, the noise equivalent of a circuit shown in Figure 1.27(a), where a 
signal source Vs is driving a hypothetical noise-free load resistor /?;„ (which can be considered an 
input of the power amplifier) through three noise resistors R lt R 2 , and 7? 3 , is a noise voltage source 
el — 2kT RjAf connected in series with an ideal (noise-free) resistor equal to the Thevenin resistance 
Rt, as shown in Figure 1.27(b). 

Consider now a simple parallel RC circuit shown in Figure 1.28(a), where the thermal noise due 
to the parallel resistor is represented by a parallel noise current source i n . Nyquist has determined the 
thermal noise output of a two-port network containing both resistive and reactive elements, as shown 
in Figure 1.28(A). In this case, the mean-square thermal noise voltage is given by 



2a/ A/ 



(1.175) 



el = IkTrAf 



(1.176) 




(1.177) 




(a) 




FIGURE 1.27 Circuit with three resistors and its equivalent with noise voltage source. 



46 



PASSIVE ELEMENTS AND CIRCUIT THEORY 




(a.) (*) 
FIGURE 1.28 Noise characterization of two-port RC network. 



O 



where integration is performed over the frequency bandwidth of interest A/ and 

R(f)= t (1-178) 

1 + (2nfCR) 2 

is the real part of the output circuit impedance at frequency/. 

Hence, the parallel current noise source can be equivalently transformed to the series noise voltage 
source by integration over infinite frequency bandwidth with the total mean-square noise voltage given 
by 

OO OO 

— 4kT f Rdoj 2kTR f doj kT 

2it J l + (a>CR) 2 it J l + (a>CR? C 
0 0 

where the resistance R has no effect on the noise voltage which depends on the value of the capacitance 
C and temperature T only [36,37]. 



1.9.2 Noise Figure 

It is well-known that any linear noisy two-port network can be represented as a noise-free two-port 
part with noise sources at the input and the output connected in different way [38,39]. For example, 
the noisy linear two-port network with internal noise sources shown in Figure 1.29(a) can be redrawn, 
either in the impedance form with external series voltage noise sources shown in Figure 1.29(b) or in 
the admittance form with external parallel current noise sources shown in Figure 1.29(c). 

However, to fully describe the noise properties of the two-port network at fixed frequency, some- 
times it is convenient to represent it through the noise-free two-port part and the noise sources 
equivalently located at the input. Such a circuit is equivalent to the configurations with noise sources 
located at the input and the output [40]. In this case, it is enough to use four parameters: the noise 
spectral densities of both noise sources and the real and imaginary parts of its correlation spectral 
density. These four parameters can be defined by measurements at the two-port network terminals. 
The two-port network current and voltage amplitudes are related to each other through a system of 
two linear algebraic equations. By taking into account the noise sources at the input and the output, 
these equations in the impedance and admittance forms can be respectively written as 



V2 — Z21/1 + Z22/2 — Vn2 



(1.180) 

(1.181) 



NOISE 47 



V| 



O 



,.,(t) 



0- 



Noisy 
two-port 
network 



(a) 



Noise-free 
impedance 

two-port 

network 



(/») 



Noise-free 
admittance 
two-port 
network 



(0 



-O 



<-'n2 | 2 



CD 



fa 



-o 



FIGURE 1.29 Linear two-port network with noise sources. 



and 

h = Y n V 1 + Y 12 V 2 -I nl (1.182) 
I 2 = YnVi +Y 22 V 2 (1.183) 

where the voltage and current noise amplitudes represent the Fourier transforms of noise fluctuations. 

The equivalent two-port network with voltage and current noise sources located at its input is 
shown in Figure 1.30(a), where [Y] is the two-port network admittance matrix and ratios between 
current and voltage amplitudes can be written as 



h = Y n (Vi + V m ) + Y 12 V 2 - /„ 

h = y 21 (Vi + vy + Y 22 V 2 . 



(1.184) 
(1.185) 



48 



PASSIVE ELEMENTS AND CIRCUIT THEORY 



V| 



-©■ 



0 



Noise- free 
two-port 
network 

m 



(a) 



i] f„, 



V| 



>.&) 



Noise-free 
two-port 
network 
[Z] 



(b) 



-O 



-O 



FIGURE 1.30 Linear two-port network with noise sources at input. 



From comparison of Eqs. (1.182) and (1.183) with Eqs. (1.184) and (1.185) it follows that 

V m = -^ (1.186) 

/m = /„! " ^-/n2 (1.187) 

representing the relationships between the current noise sources at the input and the output corre- 
sponding to the circuit shown in Figure 1.29(c) and the voltage and current noise sources at the input 
only corresponding to the circuit shown in Figure 1.30(a). In this case, Eqs. (1.186) and (1.187) 
are valid only if Y 2 i ^ 0 that always takes place in practice. Similar equations can be written for 
the circuit with the series noise voltage source followed by a parallel noise current source shown in 
Figure 1 30(b) in terms of impedance Z-parameters to represent the relationships between the voltage 
noise sources at the input and the output corresponding to the circuit shown in Figure 1.29(A). The 
use of voltage and current noise sources at the input enables the combination of all internal two-port 
network noise sources. 

To evaluate the quality of a two-port network, it is important to know the amount of noise added 
to a signal passing through it. Usually, this can be done by introducing an important parameter such 
as a noise figure or noise factor. The noise figure of the two-port network is intended as an indication 
of its noisiness. The lower the noise figure, the less is the noise contributed by the two-port network. 
The noise figure is defined as 

F — 5m/jVm (1.188) 
S M /N 

Out 

where SiJN in is the signal-to-noise ratio available at the input and S out /N OM is the signal-to-noise ratio 
available at the output. 



NOISE 49 



For a two-port network characterizing by the available power gain Ga, the noise figure can be 
rewritten as 



F = 



■VMn 



G A S m /G A (M„ + A'add) 



= 1 + 



(1.18 



where N^a is the additional noise power added by the two-port network referred to the input. From 
Eq. (1.189) it follows that the noise figure depends on the source impedance Z s shown in Figure 
1.31(a), but not on the circuit connected to the output of the two-port network. 

Hence, if the two-port network is driven from the source with impedance Z s = R s + jX$, the noise 
figure F of this two-port network in terms of the model shown in Figure 1.31 (ft) with input voltage 
and current noise sources and noise-free two-port network can be obtained by 



1 + 



K + Zs'nl 

4kTR s Af 



1 + 



R a + \Z S \ 2 G n + 2v^G;Re(CZ s ) 



(1.190) 



A 



3-0-0- 



Noisy 
two-port 
network 



(a) 



(ft) 



0 >• 0 0 
_L_ 



(0 



Noise-free 
two-port 
network 



Noise-free 
two-port 
network 



FIGURE 1.31 Linear two-port networks to calculate noise figure. 



50 PASSIVE ELEMENTS AND CIRCUIT THEORY 



where 



Rn= — (1.191) 

4kT Af 

is the equivalent input-referred noise resistance corresponding to the noise voltage source, 

p. 

G„ = - — (1.192) 

AkTAf 

is the equivalent input-referred noise conductance corresponding to the noise current source, and 



C 



(1.193) 



is the correlation coefficient representing a complex number less than or equal to unity in magnitude 
[39]. Here, G„ and R B generally do not represent the particular circuit immitances but depend on the 
bias level resulting in a dependence of the noise figure on the operating bias point of the active device. 

As the source impedance Zs is varied over all values with positive R$, the noise figure F has a 
minimum value of 



l + 2jR^ n 



(ImC) 2 + ReC 



(1.194) 



which occurs for the optimum source impedance Z Sopt = Rs opt + /X'sopt given by 



i2 



Zsopt = (1-195) 




X Sop t = J 7 ^lmC. (1.196) 

As a result, the noise figure F for the input impedance Zs which is not optimum can be expressed 
in terms of F mm as 

I |2 I" I \2 / \2~| G n 

F = F mm + Zs — Zs 0 pt| — = Fmin + (Rs ~ ^Sopl ) + l-Xs — Xsopt) — ■ (1.197) 

"S L J w s 

Similarly, the noise figure F can be equivalently expressed using a model shown in Figure 1.31(c) 
with source admittance Y s — Gs + jB$ as 



^Soptl -p- — Fmin + \{Gs — G So pt) + (B S — Ssopt) 1 ~pr 



(1.198) 



where F^ is the minimum noise figure of the two-port network which can be realized with respect 
to the source admittance Ys, l^sopt = Gsopt + jBsopt is the optimal source admittance, and R n is the 
equivalent noise resistance which measures how rapidly the noise figure degrades when the source 
admittance Ys deviates from its optimum value Fsopt [41]. Since the admittance Ys is generally 
complex, then its real and imaginary parts can be controlled independently. To obtain the minimum 
value of the noise figure, the two matching conditions of Gs = Gs op i and5 s — 5s op t must be satisfied. 

The physical interpretation of the noise sources which are assumed to be stationary random pro- 
cesses is given by their self- and cross-power spectral densities which are defined as the Fourier 



NOISE 51 



transform of their auto- and cross-correlation function. These spectral densities in two-port matrix 
form leads to the so-called correlation matrices with their admittance, impedance, or chain repre- 
sentations [42]. The correlation matrix C belonging to the noise sources s n i and s U 2 can be written 

as 



1 

a7 



(1.199) 



where the asterisk denotes the complex conjugate. For example, the admittance correlation matrix 
for the circuit shown in Figure 1.29 (c) with two parallel current noise sources is obtained as 



Cv 



1 



A/ L'n2'„*l *'»2&. 

Determination of the noise correlation matrix is based on the following procedure: 



(1.200) 



• Each element in the diagonal matrix is equal to the sum of the noise current of each element 
connected to the corresponding node: the first diagonal element is the sum of noise currents 
connected to the node 1, while the second diagonal element is the sum of noise currents 
connected to node 2. 

• The off-diagonal elements are the negative noise current of the element connected to the pair 
of the corresponding node; therefore, a noise current source between nodes 1 and 2 goes into 
the matrix at locations (1, 2) and (2, 1). 

• If a noise current source is grounded, it will only contribute to one entry in the noise correlation 
matrix — at the appropriate location on the diagonal; if it is not grounded, it will contribute 
to four entries in the matrix — two diagonal entries corresponding to the two nodes and two 
off-diagonal entries. 

By applying these rules for the circuit with two current noise sources shown in Figure 1.32, the 
admittance noise correlation matrix Cy can be defined as 



1 

A/ 



i 2 +z' 2 



(1.201) 



'nl 




FIGURE 1.32 Circuit with two noise current sources. 



52 PASSIVE ELEMENTS AND CIRCUIT THEORY 



To form the impedance noise correlation matrix with voltage noise sources, we can write 

i fnif^i = [Z] [Cy] [Z]T 



(1.202) 



where [Z] is the impedance Z-matrix of the two-port network and T denotes the Hermitian or 
transposed conjugation. 

In the case where the correlation matrix cannot be theoretically derived, the measurements can 
be used for its determination. Such measurements are usually done by defining the equivalent noise 
resistance R n , the optimal source admittance F Sop t, and the minimum noise figure F^. As a result, 
the chain representation of the noise correlation matrix is obtained as 



C A = 4kT 



1 



^n^Sopt 



^n^sopt 



Rn Y, 



n | * Sopt | 



(1.203) 



where T is the absolute temperature [42], If the correlation matrix has been determined, the noise 
parameters can be calculated analytically from 



4kT 



[Sopt 



L -22 



i + 



Im 2 



r A 

L 12 

L ii 



;'Im 



r A 

12 

r A 



C"j2 4- CfjFsopt 
2kf ' 



(1.204) 
(1.205) 
(1.206) 



where Cj\, C^, C A j, and are the elements of the chain correlation matrix Ca- 

In a multistage transmitter system, the input signal travels through a cascade of many different 
components, each of which may degrade the signal-to-noise ratio to some degree. For a cascade of 
two stages having available gains Gai and Ga2 an d noise figures Fj and F 2 , using Eq. (1.189) results 
in the output-to-input noise power ratio N out /Ni n written as 



1 + 



^adt 



+ 



GaiGa2 I Fi + 



(1.207) 



where A'addi and N^m are the additional noise powers added by the first and second stages, respectively. 
Consequently, an overall noise figure F\% for a two-stage system based on Eq. (1.188) can be given 
by 



Oai 

Eq. (1.208) can be generalized to a multistage transmitter system with n stages as 

F n -1 



F 2 - 1 

Fin = Fi + 4^ + ■ 



+ 



GaiGa 2 • • ■ Ga(ii-I) 



(1.208) 



(1.209) 



which means that the noise figure of the first stage has the predominant effect on the overall noise 
figure, unless Gai is small or F 2 is large [43]. 



REFERENCES 53 



1.9.3 Flicker Noise 

The flicker or \lf noise is a low-frequency noise associated with a fluctuation in the conductance with 
a power spectral density proportional to/ ~ y , where y = 1 .0 ± 0. 1 in a wide frequency range, usually 
measured from 1 Hz to 10 kHz [44]. Its spectrum cannot be exactly f~ l at offset frequencies from 
/ = 0 to/ —> oo, since neither the integral over the power density nor the Fourier transform would 
be able to have finite values. Unlike the thermal or shot noise sources, the origin of the 1//' noise is 
not exactly clear and open to debate despite its predictable behavior. Generally, it is a result of both 
surface and bulk effects in the semiconductor material and is not generated by the current. In series 
experiments it was shown that there is a type of 1// noise that is a fluctuation in the carrier mobility 
due to lattice scattering. 

Significant contribution to the low-frequency noise is made by the generation-recombination and 
burst noises [45]. The generation-recombination noise associated with the fluctuations in the number 
of the carriers rather than their mobility is due to trap centers within the bandgap of a semiconductor. 
It may have any frequency behavior between/ 0 and/ -2 . If not masked by thermal noise, the low- 
frequency noise generated from these trap centers becomes f~ 2 at very high frequencies. However, 
if the lifetime of the carriers in the semiconductor is finite, the noise spectral density reaches a 
plateau at very low frequencies. Burst noise (random telegraph noise) is a special kind of generation- 
recombination noise due to a single trap in the active device region. It is often observed in submicron 
devices or in devices with very poor crystalline quality. In such devices, a trap level with certain 
energy and at a specific location in the active device region (a single localized trap) traps and detraps 
the carriers causing an on-off time-dependent signal similar to a telegraph signal [46]. 

The physical origin of a low-frequency 1// noise for any type of the metal-oxide-semiconductor 
field-effect transistor (MOSFET) devices including CMOS transistors is based on two dominant 
processes: random fluctuation of the carriers in the channel due to fluctuations in the surface potential 
caused by trapping and releasing of the carriers by traps located near the Si-Si02 interface, and 
mobility fluctuations due to carrier interactions with lattice fluctuations [47]. However, for a CMOS 
transistor depending on its type, one effect can prevail over the other. For example, flicker noise 
in ^-channel devices is mostly attributed to carrier number fluctuations, while flicker noise in p- 
channel devices is often attributed to mobility fluctuations. It was observed that pMOS transistors 
have significantly lower \lf noise than nMOS transistors of the same size and fabricated with the 
same CMOS process (by one order of magnitude or more). This is because, when an n + -poly silicon 
gate layer is used for both the nMOS and pMOS devices, nMOS transistors have a surface channel 
while pMOS transistors have a buried channel [48]. 

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54 PASSIVE ELEMENTS AND CIRCUIT THEORY 



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22. W. Hilberg, "From Approximations to Exact Relations for Characteristic Impedances," IEEE Trans. Mi- 
crowave Theory Tech., vol. MTT-17, pp. 259-265, May 1969. 

23. I. J. Bahl and R. Garg, "A Designer's Guide to Stripline Circuits," Microwaves, vol. 17, pp. 90-96, Jan. 1978. 

24. D. D. Grieg and H. F. Engelmann, "Microstrip - A New Transmission Technique for the Kilomegacycle 
Range," Proc . IRE, vol. 40, pp. 1644-1650, Dec. 1952. 

25. E. O. Hammerstad, "Equations for Microstrip Circuit Design," Proc. 5th Europ. Microwave Conf., pp. 
268-272, Sept. 1975. 

26. I. J. Bahl and R. Garg, "Simple and Accurate Formulas for Microstrip with Finite Strip Thickness," Proc. 
IEEE, vol. 65, pp. 1611-1612, Nov. 1977. 

27. R. S. Carson, High-Frequency Amplifiers, New York: John Wiley & Sons, 1975. 

28. K. C. Gupta, R. Garg, and R. Chadha, Computer-Aided Design of Microwave Circuits, Dedham: Artech 
House, 1981. 

29. S. B. Cohn, "Slot Line on a Dielectric Substrate," IEEE Trans. Microwave Theory Tech., vol. MTT-17, 
pp. 768-778, Oct. 1969. 

30. E. A. Mariani, C. P. Heinzman, J. P. Agrios, and S. B. Cohn, "Slot Line Characteristics," IEEE Trans. 
Microwave Theory Tech., vol. MTT-17, pp. 1091-1096, Dec. 1969. 

31. R. Garg and K. C. Gupta, "Expression for Wavelength and Impedance of a Slotline," IEEE Trans. Microwave 
Theoiy Tech., vol. MTT-24, p. 532, Aug. 1976. 

32. C. P. Weng, "Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Nonreciprocal Gyro- 
magnetic Device Applications," IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp. 1087-1090, Dec. 
1969. 

33. R. N. Simons, Coplanar Waveguide Circuits, Components, and Systems, New York: John Wiley & Sons, 
2001. 

34. G. Ghione and C. Naldli, "Analytical Formulas for Coplanar Lines in Hybrid and Monolithic," Electronics 
Lett., vol. 20, pp. 179-181, Feb. 1984. 

35. A. van der Ziel, Noise, Englewood Cliffs: Prentice-Hall, 1954. 

36. H. L. Krauss, C. W. Bostian, and F. H. Raab, Solid State Radio Engineering, New York: John Wiley & Sons, 
1980. 



REFERENCES 55 



37. B. L. Buckingham, Noise in Electronic Devices and Systems, New York: John Wiley & Sons, 1983. 

38. H. C. Montgomery, "Transistor Noise in Circuit Applications," Proc . IRE, vol. 40, pp. 1461-1471, 1952. 

39. H. Rothe and W. Dahlke, "Theory of Noise Fourpoles," Proc. IRE, vol. 44, pp. 811-818, June 1956. 

40. A. G. T. Becking, H. Groendijk, and K. S. Knol, "The Noise Factor of Four- Terminal Networks," Philips Res. 
Rep., vol. 10, pp. 349-357, 1955. 

41. IRE Subcommittee 7.9 on Noise (H. A. Haus, Chairman), "Representation of Noise in Linear Twoports," 
Proc. IRE, vol. 48, pp. 69-74, Jan. 1960. 

42. H. Hillbrand and P. H. Russer, "An Efficient Method for Computer Aided Noise Analysis of Linear Amplifier 
Networks," IEEE Trans. Circuits and Systems, vol. CAS-23, pp. 235-238, Apr. 1976. 

43. H. T. Friis, "Noise Figures of Radio Receivers," Proc. IRE, vol. 32, pp. 419^422, July 1944. 

44. F. N. Hooge, "1// Noise Sources," IEEE Trans. Electron Devices, vol. ED-41, pp. 1926-1934, Nov. 1996. 

45. S. Mohammadi and D. Pavlidis, "A Nonfundamental Theory of Low-Frequency Noise in Semiconductor 
Devices," IEEE Trans. Electron Devices, vol. ED-47, pp. 2009-2017, Nov. 2000. 

46. X. N. Zhang, A. van der Ziel, K. H. Duh, and H. Morkoc, "Burst and Low-Frequency Generation- 
Recombination Noise in Double-Heterojunction Bipolar Transistors," IEEE Trans. Electron Device Lett., 
vol. 5, pp. 277-279, July 1984. 

47. Y. P. Tsividis, Operation and Modeling of the MOS Transistor, New York: McGraw-Hill, 1987. 

48. K. K. O, N. Park, and D.-J. Yang, "1// Noise of NMOS and PMOS Transistors and Their Implications to 
Design of Voltage Controlled Oscillators," 2002 IEEE RFIC Symp. Dig., pp. 59-62, June 2002. 



Active Devices and Modeling 



Accurate device modeling is an extremely important procedure in circuit design, especially in mono- 
lithic integrated circuits. Better approximations of the final design can only be achieved if the 
nonlinear device behavior is described accurately. Once a device model has been incorporated into 
a circuit simulator, it requires the parameters to specify the device characteristics according to the 
model equations. The next step is to provide a procedure for adequate extraction of S-parameter 
data. This poses the problem of how to extract the device parameters from the measurement results 
accurately, rapidly, and without unnecessary measurement complexity. The best way is to minimize 
the device bias points under S-parameter measurements and to combine the analytical approach with 
a final optimization procedure to provide the best fitting of the experimental curves and empirical 
equation-based model curves. Numerical optimization is often used to fit the model S-parameters 
to the measured parameters since the resulting device element values depend on the starting values 
and are not unique. The analytical approach incorporates a derivation of the basic intrinsic device 
parameters from its equivalent circuit based on S- to Y- or Z-parameter transformations using suffi- 
ciently simple equations. However, it is crucial to choose an appropriately large-signal model that is 
most suitable for a specific active device, accurately describes the nonlinear behavior of its equivalent 
circuit parameters, and contains a reasonable number of model parameters to be determined. 

This chapter describes all necessary steps for an accurate device modeling procedure, begin- 
ning with determining the small-signal equivalent circuit parameters. A variety of nonlinear models, 
including noise models, for metal-oxide-semiconductor field-effect transistors (MOSFETs), metal- 
semiconductor field-effect transistors (MESFETs), high electron mobility transistors (HEMTs), bipo- 
lar devices including heterojunction bipolar transistors (HBTs), which are very prospective to design 
modern microwave hybrid and monolithic integrated circuits, are given and discussed. 

2.1 DIODES 

Most modern diodes are based on semiconductor p-n junctions. In a p-n diode, conventional current 
can flow from the p-type side (the anode) to the ii-type side (the cathode), but cannot flow in the 
opposite direction. Another type of semiconductor diode, the Schottky diode, is formed from the 
contact between a metal and a semiconductor rather than by a p-n junction. 

2.1.1 Operation Principle 

A semiconductor diode current-voltage (/-V) characteristic curve is ascribed to the behavior of the 
so-called depletion layer or depletion zone that exists at the p-n junction between the differing 
semiconductors. When a p-n junction is first created, conduction band (mobile) electrons from the 
n-doped region diffuse into the p-doped region, where there is a large population of holes (places 
for electrons in which no electron is present) with which the electrons "recombine." When a mobile 
electron recombines with a hole, the hole vanishes and the electron is no longer mobile. Thus, two 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



57 



58 



ACTIVE DEVICES AND MODELING 



charge carriers have vanished. The region around the p-n junction becomes depleted of charge carriers 
and thus behaves as an insulator. 

However, the depletion width cannot grow without limit. For each electron-hole pair that recom- 
bines, a positively charged dopant ion is left behind in the n-doped region, and a negatively charged 
dopant ion is left behind in the p-doped region. As recombination proceeds and more ions are created, 
an increasing electric field develops through the depletion zone that acts to slow and then finally 
stops recombination. At this point, there is a built-in potential across the depletion zone. This built-in 
potential is positive because the »-side is at a higher potential than the p-side, which is proper to 
obtain a balance between drift and diffusion across the junction. 

If an external voltage is placed across the diode with the same polarity as the built-in potential, 
the depletion zone continues to act as an insulator preventing a significant electric current. This is 
the reverse bias phenomenon. However, if the polarity of the external voltage opposes the built-in 
potential, recombination can once again proceed resulting in substantial electric current through the 
p-n junction. For silicon diodes, the built-in potential is approximately 0.6 V. Thus, if an external 
current is passed through the diode, about 0.6 V will be developed across the diode such that the 
p-doped region is positive with respect to the n-doped region and the diode is said to be "turned on" 
as it has a. forward bias. 

A diode I—V characteristic shown in Figure 2.1 can be approximated by two regions of operation. 
Below a certain difference in potential between the two leads, the depletion layer has significant 
width, and the diode can be thought of as an open (nonconductive) circuit. As the potential difference 
is increased, at some stage the diode will become conductive and allow charges to flow, at which 
point it can be thought of as a connection with zero (or at least very low) resistance. In a normal 
silicon diode at rated currents, the voltage drop across a conducting diode is approximately 0.6 to 
0.7 V. In the reverse bias region for a normal p-n rectifier diode, the current through the device is 



Current 



Breakdown 
voltage 



Leakage current 



Avalanche 
current 

Reverse 

volwgc 




FIGURE 2.1 Diode current-voltage characteristic. 



DIODES 59 



very low (in the jjA range) for all reverse voltages up to a point called the peak inverse voltage (PIV). 
Beyond this point a process called reverse breakdown occurs that causes the device to be damaged 
along with a large increase in current. For special purpose diodes like the avalanche or Zener diodes, 
the concept of PIV is not applicable since they have a deliberate breakdown beyond a known reverse 
current such that the reverse voltage is "clamped" to a known value (called the Zener voltage or 
breakdown voltage). These devices however have a maximum limit to the current and power in the 
Zener or avalanche region. 

The Shockley ideal diode equation or the diode law is the current-voltage {I-V) characteristic of 
an ideal diode in either forward or reverse bias (or no bias): 



where / is the diode current, / sat is a scale factor called the reverse saturation current, V is the 
voltage across the diode, Vj is the thermal voltage, and n is the emission coefficient, also known as 
the ideality factor. The term "saturation" indicates that the current I approaches an asymptote and 
becomes independent of the voltage V. 

It is derived with the assumption that the only processes giving rise to current in the diode are 
drift (due to electrical field), diffusion, and thermal recombination-generation. It also assumes that 
the carrier recombination-generation current in the depletion region is insignificant. Additionally, 
it does not describe the "leveling off" of the I-V curve at high forward bias due to internal resis- 
tance, nor does it explain the practical deviation from the ideal at very low forward bias due to 
recombination-generation current in the depletion region. 

The emission coefficient n varies from about 1 to 2 depending on the fabrication process and 
semiconductor material and in many cases is assumed to be approximately equal to 1 (thus omitted). 
The thermal voltage Vj is approximately 25.7 mV at room temperature 25°C or 298 K defined by 



where q is the electronic charge (or elementary charge), k is Boltzmann constant, and T is the absolute 
temperature of the p—n junction. 

The diode small-signal electrical behavior can be described by the equivalent representation of 
the p-n junction with the differential resistance at the operating point 



and by two forms of charge storage: charge storage in the depletion region due to the dopant 
concentrations and charge storage due to minority-carrier charges injected into the neutral p- and 
n-type regions resulting in the junction and diffusion capacitances, respectively [1]. There are several 
types of semiconductor junction diodes, which either emphasizes a different physical aspects of a 
diode often by geometric scaling, doping level, choosing the right electrodes, or just an application 
of a diode in a special circuit. 

2.1.2 Schottky Diodes 

Schottky diodes are constructed from a structure made by forming a contact between a metal and 
a semiconductor defined as a metal-semiconductor junction. Each metal-semiconductor junction 
is characterized by a potential barrier called the Schottky barrier, which depends only on the two 
materials and temperature, and it is not a function of the semiconductor doping. The Schottky barrier 
blocks the flow of electrons from the metal toward the semiconductor. Schottky diodes have a lower 




(2.1) 




(2.2) 




(2.3) 



60 



ACTIVE DEVICES AND MODELING 



forward voltage drop than any p-n junction diode and its electrical behavior is described by similar 
Shockley diode current-voltage equation. Their forward voltage drop at forward currents of about 1 
mA is in the range 0.15 to 0.45 V, which makes them useful in voltage clamping applications and 
prevention of transistor saturation. 

Reverse-bias breakdown in Schottky diodes formed on lightly doped substrates occurs through 
avalanching in high-field regions, just as in a p-n junction diode. However, Schottky diodes tend 
to present lower breakdown voltage than p-n junction diodes. For Schottky diodes formed on more 
heavily doped substrates, breakdown may occur through tunneling, which is quite comparable to 
that of Zener breakdown in p-n junction diodes. Schottky diodes are usually referred to as majority- 
carrier devices and the forward-bias current in these devices is characterized by electron emission from 
/j-type semiconductor into the metal, rather than by hole injection into the semiconductor. Therefore, 
they do not suffer from minority-carrier storage problems that slow down most normal diodes, thus 
resulting in a faster "reverse recovery" than any p-n junction diode. This contributes toward their high 
switching speed and their suitability in high speed circuits and RF devices such as a switched-mode 
power supply, mixers, and detectors. 

Operating as a mixer diode, the Schottky diode is used as a variable-resistance diode with non- 
linear incremental small-signal conductance of the junction. The Mott diode is another variant of 
the conventional Schottky-diode structure which differs in its epitaxial layer. Being very thin and 
lightly doped, the epilayer is fully depleted and the depletion region extends into the buffer layer 
located between the epilayer and the substrate. As a result, the junction capacitance consists of two 
components that can be modeled by two capacitances connected in series: the capacitance of the 
depleted epilayer, which is very small and fixed, and the larger voltage-variable capacitance, arising 
from the part of the depletion region that extends into the buffer layer. But because the smaller fixed 
capacitance dominates, the Mott-diode capacitance, therefore, is only weakly dependent on voltage. 
The nonlinear Mott-diode model is shown in Figure 2.2, which includes the current source I(V), two 
junction capacitances in series, C and Cj, and series parasitic resistance R s [2]. Because of the Mott 
diode lower doping density, its reverse saturation current / sat is lower than that of a conventional 
Schottky diode having the same diameter. The cutoff frequency f c for both Mott and Schottky diodes 
is defined as 



where the junction capacitance Cj(0) defined at zero bias voltage is calculated at the operating bias 
point and the series resistance includes skin effect. 



1 



InR.Cj (0) 



(2.4) 



Q 




FIGURE 2.2 Mott diode equivalent circuit. 



DIODES 



61 



The most important difference between p-n junction diode and Schottky diode is reverse recovery 
time, when the diode switches from nonconducting to conducting state and vice versa. Where in p-n 
junction diode the reverse recovery time can be in the order of hundreds of nanoseconds, and less 
than 100 ns for fast diodes, Schottky diodes do not have a recovery time, because there is nothing 
to recover from. The switching time is of about 100 ps for the small signal diodes, up to tens of 
nanoseconds for special high-capacity power diodes. With p-n junction switching, there is a reverse 
recovery current associated with it, which among other things with high-power semiconductors brings 
increased electromagnetic interference (EMI) noise. With Schottky diodes switching instantly with 
only slight capacitive loading, this is of much lesser concern. 

The most evident limitations of Schottky diodes are the relatively low reverse voltage rating for 
silicon-metal Schottky diodes, 50 V and below, and a relatively high reverse leakage current. The 
reverse leakage current, increasing with temperature, leads to a thermal instability issue. This often 
limits the useful reverse voltage to well below the actual rating. Luckily, times are changing and the 
diodes are becoming better and better and the voltage ratings are now up at 200 V. The silicon carbide 
Schottky diodes have about 40 times lower reverse leakage current compared to silicon counterparts 
and are made as 300- V, 600- V, and 1200-V variants. Silicon carbide has high thermal conductivity 
and temperature has little influence on switching and thermal characteristics, plus diodes have no 
thermal runaway. With special packing it is possible to have operating junction temperatures of over 
500 K, which allows passive radiation cooling in aerospace applications. 

2.1.3 p-i-n Diodes 

A p-i-n diode is a device that operates as a variable resistor at RF and microwave frequencies. It 
is a silicon semiconductor diode in which a high-resistivity intrinsic /-layer is sandwiched between 
the p-type and n-type layers. When a p-i-n diode is forward biased, holes and electrons from p- 
and H-layers are injected into the /-layer, respectively. These charges do not recombine immediately; 
instead, a finite quantity of charge always remains stored for an average time, called the carrier- 
life time r, thus lowering the resistivity of the /-region. The quantity of stored charge Q is directly 
proportional to the carrier life time (recombination time) x and the forward bias current If as Q = Ut. 
The resistance R s of the /-layer under forward bias condition is inversely proportional to Q and directly 
proportional to W 2 , where W is the /-layer thickness. When the p-i-n diode is at zero or reverse bias, 
there is no stored charge in the /-layer and the diode appears as a junction capacitance Cj shunted 
by a parallel resistance R p . The p-i-n diode equivalent circuits for (a) forward bias and (£>) zero or 
reverse modes are shown in Figure 2.3, where L s is the parasitic series inductance [3]. By varying 



Q 




O 



6 



(a) 



6 

(b) 



FIGURE 2.3 p-i-n diode equivalent circuits. 



62 



ACTIVE DEVICES AND MODELING 



the /-layer thickness and diode area, it is possible to construct p-i-n diodes of different geometries 
to result in the same R s and C s values. These devices may have similar small-signal characteristics. 
However, a diode with thicker /-layer would have a higher breakdown voltage and better distortion 
properties. On the other hand, a diode with thinner /-layer would have faster switching speed. 

At dc and very low frequencies being forward biased, the p-i-n diode behaves similar to a p-n 
junction diode and the diode resistance is described by the dynamic (or differential) resistance of the 
diode I—V characteristic at any forward bias point. The dc dynamic resistance point is not, however, 
valid for the p-i-n diodes at frequencies above which the period is shorter than the transit time of the 
/-region. The frequency at which this occurs is called the transit time frequency fj, and this lowest 
frequency limit is primarily a function of the /-layer thickness W expressed by 

1300 

& = 1^7T MHz - (2 - 5) 

At high frequencies when a p-i-n diode is at zero and reverse bias, it appears as a parallel plate 
capacitor, essentially independent of reverse voltage being a function of the junction capacitance and 
thickness of /-layer. However, at frequencies much lower than/ T , the p-i-n diode acts as a varactor, 
and the junction capacitance Cj is determined by applying a sufficiently large reverse voltage, which 
fully depletes the /-region of carriers. At low reverse bias voltages, the finite resistivity of the /-region 
results in a lossy junction capacitance. As the reverse voltage is increased, carriers are depleted from 
the /-layer, resulting in an essentially lossless silicon capacitor. Under reverse bias, the p-i-n diode 
should not be biased beyond its dc rating voltage Vr. The diode avalanche or bulk breakdown voltage 
Vbd is proportional to the /-layer thickness W and is always higher than Vr. In a typical application, 
maximum negative voltage swing should never exceed V B d- An instantaneous excursion of the RF 
signal into the positive bias direction generally does not cause the diode to go into conduction because 
of slow reverse-to-forward switching speed. 

A step recovery diode (SRD) is a semiconductor junction diode having the ability to recover 
extremely quickly from a strong forward-conduction state to a cutoff state. It is also called the snap- 
off diode or charge-storage diode, and has a variety of uses in microwave electronic devices as pulse 
generator or parametric amplifier. The SRD must have high charge storage in the forward direction, 
low capacitance in the reverse direction, low series resistance, and high reverse breakdown voltage for 
power applications. Its switching time must be adequately short because switching speed establishes 
its high-frequency limit of operation. To meet these requirements, the SRD must have a relatively long 
charge-storage time, and the charge that is injected into the junction while it is forward-biased must 
not travel so far that it cannot be removed during the reverse-bias interval. Also, the depletion region 
must not be too wide, otherwise transit-time effects will reduce the efficiency at high frequency. 

The term "step recovery" relates to the form of the reverse recovery characteristic of these devices. 
After a forward current has been passing in an SRD and the current is interrupted or reversed, the 
reverse conduction will cease very abruptly (as in a step waveform). The SRD has the p-i-n structure, 
in which the /-region is a layer of undoped or intrinsic semiconductor or is very lightly doped [2]. 
The /-region is formed by the overlap of the p- and n-layers, both of which have a steep doping 
profile. This profile gives a narrow depletion region in thep- and «-layers and a strong built-in electric 
field, which opposes the diffusion of charge into the unction. During forward conduction, holes and 
electrons are injected into the /-layer, where they recombine very slowly. The /-layer thus becomes 
a region in which charge is stored. When the SRD is reverse-biased, the /-layer is fully depleted, 
because of the wide depletion width, which includes the entire /-layer, resulting in a very low reverse 
capacitance. Under forward bias, the SRD can be modeled as a conventional p-n junction diode, 
however, when the SRD is reverse-biased, its capacitance is nearly constant and very low. 

2.1.4 Zener Diodes 

The effect when the p-n diodes can be made to conduct backward is called the Zener breakdown. It 
occurs at a precisely defined voltage, allowing the diode to be used as a precision voltage reference. 



VARACTORS 63 




FIGURE 2.4 Circuit with Zener diode. 



A conventional p-n diode will not let significant current flow if it is reverse-biased below its reverse 
breakdown voltage. By exceeding the reverse bias breakdown voltage, a conventional diode is subject 
to high current flow due to avalanche breakdown. Unless this current is limited by external circuitry, 
the diode will be permanently damaged. A Zener diode exhibits almost the same properties, except 
the device is specially designed so as to have a greatly reduced breakdown or Zener voltage. A Zener 
diode contains a heavily doped p-n junction allowing electrons to tunnel from the valence band of 
the p-type material to the conduction band of the «-type material. A reverse-biased Zener diode will 
exhibit a controlled breakdown and let the current flow to keep the voltage across the Zener diode 
at the Zener voltage. For example, a diode with a Zener breakdown voltage of 3.2 V will exhibit a 
voltage drop of 3.2 V if reverse bias voltage applied across it is more than its Zener voltage. However, 
the current is not unlimited, so the Zener diode is typically used to generate a reference voltage for a 
power amplifier stage, or as a voltage stabilizer for low-current applications. The breakdown voltage 
can be controlled quite accurately in the doping process. Tolerances to within 0.05% are available, 
though the most widely used tolerances are 5% and 10%. 

Zener diodes are widely used to regulate the voltage across a circuit. When connected in parallel 
with a variable voltage source so that it is reverse biased, a Zener diode conducts when the voltage 
reaches the diode reverse breakdown voltage. From that point it keeps the voltage at that value. In the 
circuit shown in Figure 2.4, the bias resistor R provides the voltage drop between the input voltage 
Vi„ and the output voltage V ou t- The value of R must satisfy two conditions: it must be small enough 
that the current through diode keeps diode in reverse breakdown and it must be large enough that 
the current through diode does not destroy the device. A Zener diode used in this way is known as a 
shunt voltage regulator (shunt means connected in parallel, and voltage regulator is a class of circuit 
that produces a stable voltage across any load). 



2.2 VARACTORS 

The term "varactor" comes from the phrase "variable reactor" and means a device whose reactance 
can be varied in a controlled manner, in this case, with dc bias voltage [4]. The discovery of the 
capacitance of a rectifying contact extends back to 1929 when the first comprehensive investigation 
of this phenomenon was made [5]. In 1949 Shockley published his classic paper on the theory of p-n 
junctions where not only the expressions for the current-voltage relationships in a diode are given, 
but the capacitance effect is considered as well [6]. The possibility of tuning diodes with exponents 
greater than one-half was firstly described in 1958, with the term "hyperabrupt" coined for such 
sensitive devices [7]. 



2.2.1 Varactor Modeling 

A simplified varactor equivalent circuit is shown in Figure 2.5, where C v is the variable depletion 
layer capacitance, C p is the package capacitance, R s is the series contact and bulk resistance, L s is the 



64 



ACTIVE DEVICES AND MODELING 



c, 



a 




* 5 U 



O 



I'D 



FIGURE 2.5 Simplified varactor equivalent circuit. 



series inductance incorporating package inductance, and VD is the diode junction [8,9]. The diode is 
necessary to take into account because of the rectifying effect during a positive voltage swinging. The 
series resistance R s is a function of applied voltage and operating frequency, although in most practical 
cases it can be considered constant. Such a model neglects some parasitic linear components, which 
should be taken into account for microwave applications including distributed-line package model 
and some capacitances due to the ground proximity. However, for most high frequency applications 
up to 2.5 GHz, these parasitics would not be significant unless higher order harmonics due to the 
varactor nonlinearity affect VCO performance. 

The varactor junction capacitance C v as a function of the reverse dc bias voltage V v can be 
expressed by 



is the varactor junction sensitivity (y = 0.5 for abrupt varactors, 1 < y < 2 for hyperabrupt varactors), 
<p is the contact potential which value depends on a doping profile of varactor; and C v o = C v (0) is 
the varactor junction capacitance at V v = 0. The voltage V v is positive since it is assumed a reverse 
connection of the varactor. 

When the junction is reverse-biased, a large electric field exists in the depletion region. As the bias 
voltage is increased, this field increases to a sufficiently high value where the thermally generated 
carriers traversing the depletion layer will generate additional hole-electron pairs by collision. These 
hole-electron pairs will in turn generate additional pairs, thus causing a multiplication effect or 
avalanche of carriers. The breakdown voltage at which avalanche occurs is the upper limit for the 
varactor voltage-control range and determines the minimum varactor junction capacitance C vm j„. The 
reverse breakdown voltage is relatively insensitive to temperature variations. 

The doping concentration level in the varactor depletion region defines the difference between an 
abrupt junction varactor and a hyperabrupt junction one. For abrupt junction shown in Figure 2.6(a), 
the doping density is constant across the depletion region, whereas, for hyperabrupt junction shown 
in Figure 2.6(b), the doping density is a nonlinear function [10]. In the latter case, ion implantation 
or nonlinear epitaxial growing techniques can accomplish this. As a result, as the reverse voltage is 
increased, the higher doping density contributes to a greater capacitance change in the hyperabrupt 
varactor than in the abrupt varactor with a constant doping density. However, the averaged doping 
concentration of the undepleted epitaxial region for abrupt varactor is higher approximately by a 
factor of two, which gives the higher value of the series resistance R$ for hyperabrupt varactor. 




(2.6) 



where 




(2.7) 



VARACTORS 65 



Doping density, cm - "' 



10' 



10" 



1 0 



p 






















n 












n 





].U 



2.0 3.0 Distance, (im 

00 



Doping density, cm 



10' 



10' 



io' 5 



p 






















n 














n 





0 



1.0 



2.0 3-0 Distance, jim 

(/.) 

FIGURE 2.6 Doping density for abrupt and hyperabrupt varactors. 



The quality factor of a varactor Q v (taken into account that the varactor junction capacitance C v 
is substantially higher than the package capacitance C p ) is defined as 



Qy(V v ,C0) = 



1 



wR s C y (V v ) 



(2.8) 



being a function of operating frequency and applied voltage. Since with the increase of R s the 
varactor quality factor decreases, the Q v of the abrupt varactor is higher than that of the hyperabrupt 
varactor at low reverse bias voltage. However, at higher reverse bias voltages, the quality factor of 
the hyperabrupt varactor becomes higher due to the more rapid decrease in the hyperabrupt varactor 
capacitance. As shown in Figure 2.7, usually over the linear tuning range for reverse bias voltage 
ranging from 1 to 10 V, the <2 V for hyperabrupt varactor is lower. As a result, the output power of 
such a voltage-controlled oscillator (VCO) with hyperabrupt varactor should be lower due to higher 
power losses in the varactor. 



2.2.2 MOS Varactor 



It is well known that an MOS transistor can be used as a variable capacitance due to the dependence 
of its total charge (representing the charge on the gate, effective interface charge, and charge in the 



66 



ACTIVE DEVICES AND MODELING 




semiconductor under the oxide) on the applied voltage between bulk and gate terminals [11]. In the 
case of pMOS varactor with the source, drain, and bulk terminals together connected, an inversion 
channel with mobile holes will be realized for gate-bulk voltages V g b greater than the threshold 
voltage of the MOS transistor V t h- Further increase in V g b results in a strong inversion of the MOS 
transistor operation. On the other hand, for voltages V g b lower than Vq>, an MOS transistor operates 
in accumulation region, where the voltage at the interface between gate oxide and semiconductor is 
positive and high enough to allow electrons to move freely. 

Figure 2.8(a) shows the cross-section of the MOS transistor, where the movement behavior of the 
majority-charge carriers in the inversion, depletion, and accumulation regions is also shown [12], A 
metal oxide structure is build on the top of a lightly doped H-well diffusion layer with the gate and 
the two n + contacts inside n-well. The device bias-dependent capacitance C can be modeled as oxide 
capacitance C ox in series with the parallel connection of the capacitance Cb owing to the depletion 
region charge and C\ owing to the inversion layer charge at the gate-oxide interface. If Cb or C\ 
dominates, the MOS transistor is operated in a depletion or strong-inversion region, respectively. 
Otherwise, if neither capacitance dominates, the MOS transistor is operated in a weak-inversion 
region. The overall behavior of the MOS capacitance C versus gate-bulk voltage V g \, is qualitatively 
shown in Figure 2.8(b), where the charge carrier movement in strong- and moderate-inversion regions 
is indicated by solid lines shown in Figure 2.8(a). 

When V g approaches V t h in the moderate-inversion region, the concentration of holes at the 
oxide interface decreases steadily, but C\ continues to be much larger than Cb- However, when the 
MOS transistor enters the weak-inversion region, modulation of the depletion region becomes of 
the same importance as hole injection when d ^ Cb, being dominating effect in the depletion 
region when C; Cb. Thus, both in the strong-inversion and accumulation region, the overall device 
capacitance C approaches the oxide capacitance C ox . Physically, an abundance of holes exists at high 
Vgb immediately below the oxide and provides bottom plate of the oxide capacitor, just as abundance 
of electrons provided that plate in the case of accumulation. In this case, the parasitic resistance is 
associated with the resistive losses of electrons moving from the bulk contact to the interface between 
the bulk and depletion regions, as shown in Figure 2.8(a) by dashed lines. Such a parasitic resistance 
can be reduced by using scaled technology with shorter device gate length L [ 1 3] . With the technology 
scaling, the oxide thickness is also reduced with correspondent increase in the oxide capacitance. 
Ideally, this should result not only in the better quality factor of a MOS varactor, but also in a wider 
tuning range since a minimum depletion capacitance increases at a lower rate. 

To obtain monotonic dependence for C, it is necessary to provide the device operation with- 
out entering the accumulation region for a very wide range of gate voltage values. This can be 



VARACTORS 67 




Accumulation Depletion Weak Moderate Strong inversion y 
inversion inversion 
(*) 

FIGURE 2.8 MOS varactor and its voltage-capacitance dependence. 



accomplished by removing the connection between drain and source with bulk, by connecting the 
bulk terminal to the supply voltage as the highest dc voltage in the circuit. Figure 2.9 shows the 
voltage-capacitance dependence of such an inversion-mode MOS varactor operating in the strong-, 
moderate-, or weak-inversion regions only. However, a more attractive approach is to use the pMOS 
device in the depletion and accumulation regions only, resulting in a wider tuning range and better 
quality factor due to lower parasitic resistance since the electrons have mobility approximately three 
times higher than holes [14]. 

The voltage-capacitance dependence of such an accumulation-mode MOS varactor operating in 
the depletion and accumulation regions only is shown in Figure 2. 10(a). To realize an accumulation- 
mode MOS varactor, the formation of the strong-, moderate-, and weak-inversion regions must be 
inhibited that requires the suppression of any injection of holes in the MOS channel. This can be 
accomplished by removing the source and drain diffusion p + -doped layers and implementing the 
bulk n + -doped contacts instead of them, thus minimizing the parasitic-well resistance, as shown in 
Figure 2. 10(b). 

Since physical models describing the behavior of a MOS device in the accumulation and depletion 
regions are different, it normally comprises separate models for these regions, the integration of which 
into a common circuit simulator such as SPICE is complicated. Figure 2. 1 1(a) shows the cross-section 
of an accumulation-mode MOS varactor, where bulk represents shorted n-well contacts. Its single 
equivalent circuit is shown in Figure 2. 11 (ft), where Cf represents the fringing capacitance mainly 
associated with sidewall of the gate; L g and R p0 ] y are the parasitic inductance and resistance of the 



68 



ACTIVE DEVICES AND MODELING 



i 




C, 



0 l : V 

FIGURE 2.9 Inversion-mode MOS varactor tuning curve. 



gate electrode, respectively [15]. The resistances /f we u, ^sub and capacitances C SU bi, C su b2 are the 
substrate-related components. The resistance R S i represents the source/drain regions. The channel 
resistance R ch is the only bias-dependent resistance. It can be modeled as R cb — R s + R acc /R p , where R s 
is the bias-independent «-well resistance between n + contact and accumulation or depletion regions 
underneath the gate, R acc is the bias-dependent resistance of the accumulation layer, and R v is the 
effective resistance along the edge of the depletion region. 



C 





• £ max 


* mm 





0 K^V 

(a) 

Bulk Gale Bulk 



v n-well j 



p-subslrate 
(A) 

FIGURE 2.10 Accumulation-mode varactor and its tuning curve. 



VARACTORS 69 



Gate 
O 



Bulk 
O 



l)cplctii»i .'i accumulation I 



fc 



/j-well 

I ViiIcii. ii lava 

/>substrate 



(a) 




W O S/D 



FIGURE 2.11 MOS varactor equivalent circuit. 



In the accumulation region, R dCC becomes much smaller than R p , and R^ is approximately equal 
to R s + R acc . In the depletion region, R. dCC can be considered infinite, and R ch approaches a constant 
value of R s + R p . The gate bias-dependence model of R acc based on the measurement results can be 
empirically given by 



1 



^acc(Vgb) 
flacc(Vgb) = °0 



1 



^acc ^gb — ^dep 



for Vg b > V iep 
elsewhere 



(2.9) 
(2.10) 



where ^ acc is the fitting parameter related to the device geometry and mobility of electrons in the 
accumulation region, and Vdep is the fitting parameter related to the flat-band voltage in the depletion 
region. 

Accurate model of the MOS varactor capacitance C based on the description of its bias-dependent 
behavior in two regions separately, available in SPICE simulator and valid under different bias 



70 



ACTIVE DEVICES AND MODELING 



conditions for a frequency range up to 10 GHz, is given by 



C(V g ) 




forV gb < F C V S 



(2.11) 



1 - F c (1 + Mj) + Mj 




for Vgb > F c Vj 



C(V e ) 



(1 - F c ) 



l-Mj 



(2.12) 



where Cj, Vj, F c , and M s are the model-fitting parameters [16]. 

It should be noted that, due to the higher average doping beneath the gate and enhanced parasitic 
interconnect capacitance in the device structure representing the more parallel connected segments 
for the same gate area and smaller gate length L, the minimum capacitance will increase resulting 
in a lower capacitance ratio [17]. However, the minimum gate-length devices have the highest 
minimum quality factors since the polysilicon gate resistance at lower L dominates the overall 
parasitic resistance, which is several magnitudes smaller than the «-well resistance at large L. As a 
result, there is a tradeoff between the quality factor and capacitance tuning ratio that can be achieved, 
for example, for medium 0.65-jJ.m gate-length MOS varactors. 



2.3 MOSFETs 

Personal wireless communication services have been driving the development of silicon MOSFET 
worldwide to provide reliable low-cost and high-performance technology. For example, the laterally 
diffused MOSFET (LDMOSFET) device structures have proven to be highly efficient, high gain, 
and linear for both high-power and low-voltage microwave and RF applications, including power 
amplifiers, low-noise amplifiers, mixers, and voltage-controlled oscillators. To develop low-cost 
silicon integrated circuits using CMOS technology for higher speed and higher frequency integrated 
circuits and subsystems within shorter design time, it is necessary to create accurate device models to 
allow efficient CAD simulation. Several well-known physically based MOSFET models can describe 
the device electrical behavior [1,18]. However, some of them such as Level 1, Level 2, or Level 3 
large-signal models are very simple and cannot describe the current-voltage and voltage-capacitance 
characteristics with acceptable accuracy. Other popular models, as the BSIM3v3 or higher version 
models, are too much complicated and quite formal in general so that, for better learning of the device 
basic properties, it is useful to consider its intrinsic nonlinear core circuit only. Also, BSIM3v3 may 
not be as accurate for RFIC simulation due to their derivative discontinuity. Moreover, microwave 
parasitic effects in silicon MOSFET are not easy physically predictable. Table-based models, such as 
the HP Root model, are only accurate for the characterized structures and measurement conditions. 
An empirical analytical modeling approach is an explicit and valid compromise between physical 
models and data-based models. 

2.3.1 Small-Signal Equivalent Circuit 

To describe accurately the nonlinear properties of the large MOSFET devices, it is necessary to take 
into account the distributed nature of the gate capacitor, because the channel resistance is not equal 
to zero. In this case, the channel of such a device can be modeled as a bias-dependent RC distributed 
transmission line along the channel length, as shown in Figure 2.12. This one-dimensional approach 
assumes a gradual channel approximation when the quantity of charge in the channel is controlled 
completely by the gate electrode, only fields in the vertical dimension influence the depletion region, 
and channel current is provided entirely by drift with a constant mobility. Despite some drawbacks 
related to short-channel devices, this approach allows a compromise between accuracy and simplicity 
of a model derivation. 



MOSFETs 71 



1 — 1 1 , , , _J 


1 1 


J 


n 

\ 




l<-4 i ... =l ...i =u 

p* threshold adjust 


n~ 


n* 

\ 


i 




p epitaxial layer 




p substrate 



FIGURE 2.12 Schematic representation of MOSFET distributed channel structure. 



The ABCZ)-matrix of the given RC transmission line can be written as 



[ABCD] 



cosh yL Z 0 sinh yL 
sinh yL 



L z 0 



cosh yL 



(2.13) 



where y — JJg) R'^C'^ is the propagation constant, Z 0 = R' ch /y is the characteristic transmission line 
impedance, L is the channel length, R' ch — R ch /L, C g — C g /L,R ch is the channel charging resistance, 
which is a result of noninstantaneous respond to the changes of the gate-source voltage, and C g is the 
total gate capacitance. In this case, the equivalent gate-source impedance Z gs can be written using 
Eq. (2.13) as 



A 
C 



coth yL 

yL 



(2.14) 



The first-order approximation of Z gs obtained from a power series expansion of Eq. (2.14) yields 



Zoe R ( 



coth yL ^ R 



yL 



1 yL 
yL \yL 3 



_ ^ch 1 

3 jeoC g ' 



(2.15) 



As we see from Eq. (2.15), the MOSFET intrinsic gate-source circuit can be modeled us- 
ing a simple series circuit with the resistance R gs — R C h/3 and the capacitance C gs = C g . Fig- 
ure 2.13 shows the intrinsic transistor equivalent circuit corresponding to the first-order channel 
approximation. 

For a high-power MOSFET device whose channel width is significantly larger than its channel 
length, the distributed nature of the total gate resistance R lot — R sh W/L across the width W (where 
R S h is the sheet resistance of the gate material) has to be taken into consideration. In this case, silicon 
MOSFET can be decomposed into n devices, each with a width of Win and a gate resistance of 
R m ln. For n — »■ oo, it will be viewed as array of the small transistors distributed along the gate of 
the device. Commonly, it is necessary to consider a two-dimensional power MOSFET distributed 
model because it shows a distributed-gate nature along both the channel length and channel width. 
However, for a short-channel MOSFET, the distributed-gate effect along the channel length can be 
taken into account only in the frequency range close to the transition frequency / T and higher. When 
coR g sCgs <3C 1, an analysis of the distributed-gate model along the channel width based on transmission 



72 



ACTIVE DEVICES AND MODELING 




O d 



FIGURE 2.13 First-order approximation of intrinsic MOSFET equivalent circuit. 



line theory shows that all transistor y-parameters should be modified by the term tanh (yW)l(yW) 
[19]. However, a linear power series expansion of this term 



tanh(y W) 
yW 



= i - i^Co 



1 



1 + jcoC g 



(2.16) 



leads to only the additional use of a series lumped gate resistance ^? tot /3 that does not alter the structure 
of the transistor equivalent circuit. Consequently, the overall gate resistance R g can generally be 
divided in two consecutive series resistances as R g — R ge + R gi , where R gt is the extrinsic contact and 
ohmic gate electrode resistance and i? gi = R tot /3 is the intrinsic gate resistance due to the distributed- 
gate structure of the power MOSFET. 

The complete small-signal MOSFET equivalent circuit with the extrinsic parasitic elements is 
shown in Figure 2.14. Here, R As is the differential channel resistance as a result of the channel length 
modulation by the drain voltage, Cj s is the drain-source capacitance, L g is the gate lead inductance, 
R s and L s are the source bulk and ohmic resistance and lead inductance, R A and L d are the drain bulk 
and ohmic resistance and lead inductance, C gp and Cd P are the gate and source pad capacitances, 
respectively. 



S o 



Intrinsic 




is 



dp - 



-O d 



-O s 



FIGURE 2.14 Nonlinear MOSFET equivalent circuit with extrinsic linear elements. 



MOSFETs 73 



To characterize the transistor nonlinear electrical properties, it is sufficient to use the grounded- 
source intrinsic F-parameters. Their two-port admittance matrix is 



Y = 



1 + j^i 
g m exp(-jo)r) 



1 + j(DT g 



j(l)C gd 



-jmC gA 



+ jeo (C ds + C gd ) 



(2.17) 



where r is the effective channel carrier transit time and r g = R gs Cg S . The intrinsic gate resistance 
R gi can be considered an external gate element. In this case, the MOSFET intrinsic 7-matrix is the 
same as for the MESFET or HEMT devices. Consequently, to determine the elements of the intrinsic 
MOSFET small-signal equivalent circuit, it is possible to use similar analytical approach, which 
allows the determination of its elements through the real and imaginary parts of the device intrinsic 
admittance F-parameters. 



2.3.2 Nonlinear I-V Models 

An empirical approach to approximate the nonlinear behavior of the drain current source / ds (V gs , 
Vds) °f a JFET device is described in [20]. Instead of using separate equations for the triode and 
pinch-off regions, irrespective of the device geometry and material parameters, a general expression 
based on hyperbolic functions was proposed: 



/rls — lit 



r. 



tanhff 



(2.18) 



where /d SS is the saturation drain current for V gs = 0, a is the saturation voltage parameter, and 
V p is the pinch-off voltage. As a result, good agreement was obtained between the predicted and 
experimental results, which showed a promising prospect of such a simple empirical model. 

A similar and sufficiently simple nonlinear model using a hyperbolic function and incorporating 
self-heating effect was later proposed to describe the I-V characteristics of a MOSFET device: 



/ds = ^ff^ g I t Gexp (l + Al/ ds )tanh 



aV, 



Js 



^SATexp 



gsl 



(2.19) 



where 



K s , = V st ln 



1 + exp 



"gstl 



yV ds 



where GMexp, SATexp, and fi clit are the channel current parameters [21]. 

An empirical nonlinear model, which is single-piece and continuously differentiable, developed 
for silicon LDMOS transistors is given by 



p (1 + XVis) tanh ( [1 + K , exp (V B Reffi)] + h s exp ( Vis y J m j (2.20) 



74 



ACTIVE DEVICES AND MODELING 



where 



Kst = 14, In 



V g st2 — V gst i 



1 + exp 



Vg*2 



\ (v gstl + J(V gstl -V K ) 2 + Ai - Jvt + A^ 



Vgsti = V gs - V th0 - yV is 

Vi s — ^BReff , ,. Vda 

VBReffl = + Mi 

^2 VBReff 

^BReff = ~Y [1 + tanh ( M l " ^gstM 2 )] 

where A is the drain current slope parameter, B is the transconductance parameter, V t ho is the forward 
threshold voltage, V st is the subthreshold slope coefficient, Vj is the temperature voltage, / ss is the 
forward diode leakage current, Vbr is the breakdown voltage, K\, K2, Mi, M%, and M3 are the 
breakdown parameters, Vk, VGexp, A, and y are the gate-source voltage parameters [22]. 

In many applications, it is necessary to take into consideration the MOSFET operation in the 
weak-inversion region when the gate-source voltage V gs is smaller than the threshold voltage V&. For 
example, to improve the conversion gain of a mixer or reduce the intermodulation distortion (IMD) of 
a class AB power amplifier when device is biased around the onset of the strong-inversion region from 
the weak-inversion region for low drain quiescent current. The drain current in the weak-inversion 
region is mainly dominated by the diffusion component that increases exponentially with the gate 
voltage [11]. On the other hand, in the strong-inversion saturation region when the gate-source voltage 
is greater than the threshold voltage, the drain current is proportional to the square of (V gs - V&). 

To obtain continuous behavior from weak-inversion region to strong-inversion region for the drain 
current and a compromise between accurate device modeling and ease of circuit analysis, we can use 

/* {V gs ) = A {In [1 + exp (5 (V gs - V th ))]} 2 (2.21) 

where A and B are the approximation parameters. In this case, the drain current is effectively 
proportional to the square of (V gs - Vth) when V gs is larger than V tb and exponentially decreases with 
the gate-source voltage when V gs is smaller than V th . The approximation parameters A and B are 
defined from the following conditions: 



Ids\v ss =V lb — Ah 



9/ds 

dV gs 



= St, (2.22) 

v gs =v th 



where /(h is the threshold drain current, as shown in Figure 2. 15, and 5th is aslope of the current-voltage 
transfer characteristic in the threshold point. Then, 

A=-^ B = ^ln2. (2.23) 
Qn2) 2 /a, 

Consequently, the transfer characteristic can be defined in terms of only two physical parameters 
/a, and S^, which are easily calculated from the device measurements. By using similar analytical 
approach, the EKV MOST model has been successfully applied to low- voltage and low-current analog 
circuit design and simulation, referring voltage V g , Va, and V s to the device local substrate [23]. 



MOSFETs 75 



In 4s 



In / lh 



FIGURE 2.15 Drain current versus gate-source voltage. 



In view of the monotonous behavior of 7^ — Vfa curves, the entire drain current-voltage charac- 
teristics of a MOSFET device can be described as: 

(2.24) 



where 

Anax = / S at(l + ^V ds )tanh(aV ds ) 

where I m is the saturated drain current, a is the saturation voltage parameter (which affects a slope of 
the /ds-^ds characteristics in the linear region), X is the parameter that determines a slope of the same 
drain characteristics in the saturation region, V th — V th0 — crVds, n and /S are the fitting parameters 
that determine a slope of the transfer characteristics under large values of V gs , a is the parameter that 
expresses empirically the dependence of the threshold voltage on V is [24]. 

To verify that this model is applicable not only to high-power LDMOSFET devices but also to 
low- voltage MOSFETs, the appropriate low-voltage MOSFET power device with a gate width of W — 
2 mm was selected [25] . Figure 2. 1 6(a) shows the transistor theoretical and experimental drain current 
curves /ds(^ds)- The resulting current mean-square error of a family of the output current-voltage 
7d S -V ds characteristics is 0.42%. The transfer Id S -V gs characteristics were compared with the same 
characteristics calculated from the BSIM3v3 model, which was developed for modeling of deep 
submicrometer devices. The results shown in Figure 2.16(b) demonstrate a good agreement with the 
experimental curves and practically the same as in the case of the BSIM3v3 approximation. 

2.3.3 Nonlinear C-V Models 

The input capacitance C gs normally influences the IMD level especially when the frequency increases 
in the microwave region [26] . Generally, the calculation of the gate-source capacitance C gs or the 
gate-drain capacitance C gd from the charges corresponding to the strong-inversion model only results 
in a mathematically complicated expression [11]. Therefore, in most cases when it is necessary to 




lis (Vp, V is ) = lj 



1 + 



76 



ACTIVE DEVICES AND MODELING 




predict efficiency, gain, or output power of the power amplifier or oscillator, the capacitances C gs 
and C g d can be modeled as the fixed capacitances measured at the quiescent bias voltage, and the 
p-n junction diode capacitance model can be applied to the capacitance Cd S [27]. The gate-drain 
capacitance C g d can also be considered as the bias-dependent junction capacitance [28]. With the 
increase in the drain bias voltage, a depletion region is formed under the oxide in the lightly doped drain 
region. Therefore, the capacitance C g d can be considered as a junction capacitance, which strongly 
depends on the drain-source bias voltage Vd S - According to the accurate charge model calculations, 
the gate-drain capacitance C g d has a strong dependence on V gs only in the moderate-inversion region 
when V gs - V& < 1 V [1 1]. In this region, the behavior of C g d is similar to C gs , and can be evaluated 
using the same hyperbolic tangent functions. However, for high-voltage LDMOSFET devices, since 
the dependence of C g d on V gs is quite small, it seems sufficient to limit the dependence to Vds only. 



MOSFETs 77 




The drain-source capacitance Cd S varies due to the change in the depletion region, which is mainly 
determined by the value of Vds- 

The gate-source capacitance C gs can be described as a function of the gate-source bias voltage. 
First, we consider an appropriate behavior of each of its main composite part: the intrinsic gate-source 
capacitance C gs j, including both the gate-source and the source-substrate charge fluctuations, and 
the gate-substrate capacitance C g bi- The gate-source voltage dependence of these components is 
substantially different [11]. The intrinsic gate-substrate capacitance C g bi is constant in the accumu- 
lation region where it is equal to the total intrinsic oxide capacitance C ox , slightly decreases in the 
weak-inversion region, significantly reduces in the moderate-inversion region, and becomes prac- 
tically constant in the strong-inversion or saturation region. The intrinsic gate-source capacitance 
C gS i grows rapidly in the moderate-inversion region and equals 2C ox /3 in the saturation region. The 
dependence of the total gate-source capacitance C gs as a sum of its components C gsl and C g u on V gs 
is shown in Figure 2.17. 

A hyperbolic tangent function can be used for each of two parts of the dependence C gs (V gs ), where 
the gate-source capacitance can be approximated by 



C gs = C gsmin + CA1+ tanh 



(V ss - V t ) 



(2.25) 



where C s — (C gsmax — C gsm j n )/2, C gsmax is the maximum gate-source capacitance, C gsm in is the 
minimum gate-source capacitance, S — (Si, S2) is the slope of C gs (V gs ) at each bend point V gs — 
(V s i, as shown in Figure 2.17, 



c 9C gs I 



3 Cg S 1 



(2.26) 



The total gate-source capacitance C gs as a function of V gs can be described by 



Cg S — C gS max Cgso { 1 -(- tanh 



^(^ gs - Vsi)JJ x jl + tanh 
where C gsm ax = C ox and C gso is the model fitting parameter [24] . 



(Vg. - v s2 ) 



(2.27) 



78 ACTIVE DEVICES AND MODELING 



The approximation function for the gate-source capacitance C gs as the dependence of V gs can also 
be expressed by using the two components, both containing the hyperbolic functions: 

C gs = C gsl + C gs2 { 1 + tanh [C gs6 (V gs + C gs3 )]} + C gs4 [l - tanh (C gs5 V gs )] (2.28) 

where C gs i, C gS 2, C gS 3, C gS 4, C gS 5, and C gS 6 are the approximation parameters [22], 

If we consider the dependence of the gate-source capacitance C gs on V ds for submicrometer 
MOSFET devices when C gs slightly increases with the increase of V As , the approximation expression 
for C gs as a function of both V gs and V ds can be written as 

C gs = C gs0 + C gsl { A s + 5 s tanh [C, (V gs - V th )]} 

x {D s + £ s [1 + tanh (V gs - V ds )] tanh [F s V ds - G s V gs ]} (2.29) 

where C gs o is the bias-dependent capacitance, Va, is the threshold voltage, C gs i is the scaling factor, 
and A s , 7? s , C s , D s , E s , F s , and G s are the model fitting parameters [29]. 

On the other hand, for high- voltage MOSFET devices, the dependencies of gate-drain capacitance 
C gd and drain-source capacitance Cds on can be accurately evaluated by the junction capacitance 
model as 

C gd (d S ) = C gd0 ( ds0 ) ^ + ^ (2.30) 

where m (mi for C gd or m% for C ds ) is the junction sensitivity depending on a doping profile (m =1/3 
for the linearly graded junction, m — 1/2 for the abrupt junction, and in > 1/2 for the hyperabrupt 
junction), <p is the contact potential, C gdo and C dso are the junction capacitances when V As — V Aso . For 
practical profiles of the junction, which are neither exactly abrupt nor exactly linearly graded, one 
often chooses the parameters m and tp to obtain the best matching between the theoretical model and 
the measurements. 

For submicrometer MOSFET devices, to take into account the dependence of C gd both on V gs and 
Vi S , the approximation expression for the C gd is written as 

C gd = C gd0 + C gdl {A d + S d tanh [C d {D A V gs - V As ) - V th ]} (2.31) 

where C gd o is the bias-dependent capacitance, V th is the threshold voltage, C gd i is the scaling factor, 
while Ad, B A , Cd, and £) d are the model fitting parameters [29]. 



2.3.4 Charge Conservation 

To describe the small- and large-signal device models, it is necessary to satisfy the charge conservation 
condition. For a three-terminal MOSFET device, the matrix equation for the small-signal charging 
circuit in frequency domain is given by 



~h~ 




Cgg 


-Cgd 


-Cgs" 




>g" 




h 


= ja> 


-C dg 


c dd 


— Cds 




v A 


(2.32) 


h 




-c sg 


— c sd 


Css 




v„ 





where I g , I A , and / s are the terminal current amplitudes, V g , V A , and V s are the terminal voltage 
amplitudes, and the capacitance between any two device terminals (k, 1) is described as Cy — 
dQJdV\ [11]. To transform a three-terminal device into a two-port network with a common source 
terminal, the current and voltage terminal conditions of 7 g = 7 gs , 7d = 7d S , I s — — (7 gs + 7 ds ), 



MOSFETs 79 



V s = V gs , and V& — V s = V c \ s should be taken into account. In addition, the following 



relationships between the terminal capacitances for three-terminal devices are valid: 



r 

k- it 



Cgd ~\~ Cg s — Cdg ~t~ C 



Cda — Cdg + Cj s = Cgd + C S d 

Css — ^*sg ~t~ C s d — Cgs ~t~ Cds • 

The admittance F c -matrix for such a capacitive two-port network is 



(2.33) 



jw (C gs + C gd ) -;'«Cgd 

-ja> (Cgd + C m ) jw (C ds + C m + Cgd) 



(2.34) 



where C m — Cd g — C g d is the transcapacitance, C g d represents the effect of the drain on the gate, and 
Cd g represents the effect of the gate on the drain, and these effects are different [11], Similarly to the 
I-V characteristics, there is no reason to expect that the effect of the drain voltage on the gate current, 
which is zero assuming no leakage current, is the same as the effect of the gate voltage on the drain 
current, which is significantly large. 

Therefore, for power MOSFET devices, because the transcapacitance C m is substantially less than 
C gs , it can be translated to an additional delay time r c in a frequency range up to/ T by its combining 
with the transconductance g m according to 



jcoC m = g m J 1 + 



CDj C g 



exp 



-J tan 



C m 

COj C gs 



gmexp(-ja}r c ) 



(2.35) 



where t c = C m /(&>rCgs). To satisfy the charge conservation condition, the total delay time r, shown 
in the MOSFET equivalent circuit in Figure 2.14, represents both the ideal transit time and delay 
time due to the transcapacitance. The transcapacitance C m can be easily added to the drain-source 
capacitance Cd S under parameter extraction procedure. 



2.3.5 Gate-Source Resistance 

The gate-source resistance R gs is determined by the effect of the channel inertia in responding to 
rapid changes of the time varying gate-source voltage, and varies in such a manner that the charging 
time r g — RgsCg S remains approximately constant. Thus, the increase of R gs in the velocity satura- 
tion region (when the channel conductivity decreases) is partially compensated by the decrease of 
Cgs due to nonuniform channel charge distribution [30]. The effect of R gs becomes significant at 
higher frequencies close to the transition frequency fj of the MOSFET and may not be taken into 
account when designing RF circuits operating below 2 GHz, as used for commercial wireless appli- 
cations [25,3 1]. For example, for the MOSFET with the depletion region doping concentration value 
N A — 1700 (xm -3 , the phase of the small-signal transconductance g m near/ T reaches the value only 
of-15° [11]. 



2.3.6 Temperature Dependence 

Silicon MOSFET devices are very sensitive to the operation temperature T and their characteristics 
are strongly temperature dependent [11]. The main parameters responsible for this are the effective 
carrier mobility jU and the threshold voltage V m , resulting in the increase of the drain current through 
V t h(T) and the decrease of the drain current through /n (T) with temperature. Increasing temperature 
decreases the slope of the /ds(^gs) curves. A certain value of V gs can be found, at which the drain 
current becomes practically temperature independent over a large temperature range. The variation 



80 



ACTIVE DEVICES AND MODELING 



of Vfl, with temperature in a wide range from —50 to +200° C represents a nonlinear function, which 
is slowly decreased with temperature and can be approximated by 

Vth (T) = Va, (r nom ) + Vti AT + V T2 AT 2 (2.36) 

where AT — T — T nom , T nom — 300 K (27°C), and Vti and Vx2 are the linear and quadratic temperature 
coefficients for threshold voltage [27]. 

The variation of fi with temperature can be taken into account by introducing the appropriate 
temperature variation of / sat in Eq. (2.24). The temperature variation of / sat represents an almost 
straight line, which decreases with temperature [27]. The temperature dependence I sll t(T) can be 
approximated by the linear function as 

/ S a,(r) = / S a,(r nom ) + / T Ar (2.37) 

where Ij is the linear temperature coefficient for the saturation current. 

The temperature dependencies of the MOSFET capacitances and series resistances can be de- 
scribed by the following linear equations [1,32]: 

C (T) = C (T nom ) + C T AT (2.38) 
R(T) = R(T Dom ) + R T AT (2.39) 

where C — (C gs , Cd S , C g d), R — (R g , R s , R<i), and Rj and Cj are the linear temperature coefficients 
for the capacitances and resistances, respectively. 

At high value of V gs and Vds under dc measurement, the slope of /d S - Vds curves can be negative 
that occurs due to the self-heating effect in a highly dissipated power region. For the drain current 
model given by Eq. (2.19), this effect can be taken into account by adding a linear component 
describing the temperature dependence as 

P(T) = /3 (T nom ) + PjATj (2.40) 
Y(T) = y (r nom ) + j/tATj (2.41) 

where ATj = Rt^Pus + AT, i? t h(°C/W) is the thermal resistance, Pdis is the dc power consumption 
in watts caused by dc biasing, and /S T and y T are the linear temperature coefficients with negative 
values, respectively [21]. 

Another way of taking into account the effect of the negative conductance at high biasing is to 
write the nonlinear / ds - Vds model as follows: 

lis (T) 

/d S (T, p T ) = (2.42) 

where the drain current source Ii S (J) is given by Eq. (2.24), Vd = Vds/V 1 + (wtu,) 2 , p T is the self- 
heating temperature coefficient, Vd S is the drain-source supply voltage, r th = -R t hC t h is the thermal 
time constant, R t h is the thermal resistance, C t h is the thermal capacitance, and 7j is the function 
of ambient temperature T and p T . A thermal equivalent circuit can be added to the large-signal 
MOSFET model as a parallel -R t iiC t h circuit [22]. The thermal resistance R tii can be extracted from the 
temperature measurement of the dc characteristics. Since the slope of the dc measured /d S (V / ds) curves 
changes its sign from positive to negative, the temperature coefficient p T can be evaluated under the 
condition of 

d/ds (T, bt) 

; = 0 (2.43) 

dVi. 



MOSFETs 81 



As a result, 

1 ^ ds(r) n aa\ 

Ii(T) dV is 

where /ds(^ds) curves are determined by measurement of the pulsed /dsC^ds) curves at ambient 
temperature T, and the value of I<j S (T) is fixed the same as for zero slope of I,i s (T, p T ). 

The thermal time constant r th can be extracted by comparing pulsed /d S (Vd S ) curves calculated 
under different pulse widths and duty factors. A plot of 7ds as a function of pulse width under the 
fixed gate-source and drain-source bias voltages gives an appropriate value of r th . 



2.3.7 Noise Model 

The noise behavior of the MOSFET device can be described based on its equivalent circuit rep- 
resentation, which includes the main elements responsible for the device electrical behavior and 
noise sources. The noise generated by a circuit element can be modeled as a result of a small-signal 
electrical excitation. Each noise source is considered as statistically uncorrelated to the other noise 
sources in the circuit and the contribution of each noise source to the total noise is determined on 
the individual basis. The total noise represents the root-mean-square sum of these individual noise 
contributions. Since a device channel material is resistive, it exhibits thermal noise as a major source 
of noise, which can be represented by a noise current source connected between the drain and 
the source in the MOSFET small-signal equivalent circuit shown in Figure 2.18(a), where the flicker 




Noise-free two-port network 



5 CD C 4=X CD <** 



o d 



0 
s 



(6) 



FIGURE 2.18 MOSFET equivalent circuits with noise sources. 



82 



ACTIVE DEVICES AND MODELING 



noise is also included. The induced gate current noise is modeled by the gate noise current source i% 
connected across the gate-source capacitance C gs . 

The noise voltage and current sources can be given through their mean-square values as 



4kTR g Af (2.45) 

(2.46) 



4kTrj (ft>Cgs) 2 A/ 



4kTAf 



(2.47) 



4kTR d Af (2.48) 
4kTv /,f F 

^Ay+^p-^-A/ (2.49) 



where is the induced gate noise coefficient, g m is the device transconductance, / d is the drain bias 
current, A F is the flicker noise exponent, K F is the flicker noise coefficient, C ox is the oxide capacitance 
per unit area, L e ff is the effective channel length, i? ds o is the differential drain-source resistance at 
y ds = 0, y is the channel noise coefficient [1,33]. In the long-channel devices, i? ds0 = l/g m and y — 
2/3, while both R is0 and y are complicated functions of the device parameters in the short-channel 
MOSFETs [34]. The equation for (with excluded flicker noise) valid for both short-channel and 
long-channel devices expressed through the device parameters can be written in a simple form as 

& = 4kT/U d Af (2.50) 

where 

£ - _L_ _|_ tt2y dsal 

" Vd Sat 3 (y gs _ V(h f 



Vdsat is the saturation drain-source voltage, and a is the bulk-charge effect coefficient [35]. 

The required minimum noise figure F m i n , noise resistance R n , and optimum source admittance 
F So pt using the noise correlation Ca -matrix parameters as functions of the input-referred noise voltage 
i;^, noise current (J, and the parameters of the simplified noise-free two-port network shown in 
Figure 2.18(b) can be approximately estimated by 

R n 



111 



\hJ\ J R g 




(2.51) 
(2.52) 
(2.53) 



where/ is the operation frequency and/ T = g m /27r(C gs + C gd ) [35]. 

There are two major theories to explain the physical origin of \lf noise in MOSFET devices, 
one is based on the carrier number fluctuation theory when the flicker noise is attributed to the 
random trapping and detrapping processes of charges near the Si-Si02 interface, the other is based on 
mobility fluctuation theory considering the flicker noise as a result of the fluctuations in bulk mobility 



MESFETs AND HEMTs 



83 



[36-38]. Assuming that the channel can exchange charges with the oxide traps through tunneling, the 
charge fluctuation results in fluctuation of the surface potential, which in turn modulates the channel 
carrier density. At the same time, it is considered that the fluctuation of bulk mobility is induced by 
fluctuations in phonon population through phonon scattering. Generally, the measured noise power 
in MOSFET devices has a more complicated dependence on the gate bias and oxide thickness than 
due to the predictions based on the number or bulk mobility fluctuation theory only. Also, the surface 
mobility fluctuation mechanism should be taken into account attributed to the scattering effect of 
fluctuating oxide charge [39]. The implementation of oxide and interface trapping noise into a partial 
differential equation-based semiconductor device simulator shows the correct prediction of 1//' noise 
spectral densities for submicrometer MOSFET devices operating in subthreshold and strong inversion 
in saturation [40]. 

The dependence of flicker noise power on gate bias and oxide thickness for n-channel MOSFET 
in terms of the equivalent gate noise power eLcan be modeled by the following empirical expression: 



where V gs is the gate-source voltage, V& is the threshold voltage, L is the channel length, W is the 
channel width, C ox is the gate-oxide capacitance per unit area, q is the electron charge, Ki and K 2 
are empirical constants, and m — 0.7-1.2 [38]. In this case, it was proposed that the term with Ki 
represents the contribution from the mobility fluctuation, whereas the term with Ki represents the 
contribution from the number fluctuation. 

Generally, the noise behavior of the //-channel andp-channel MOSFET devices is different, since 
the pMOS transistor being less noisy usually has a channel at a larger distance from the interface [41]. 
Unlike MOSFET device, the flicker noise of its JFET counterpart is negligible. At low temperatures 
the noise spectrum of JFET indicates the presence of the several types of the generation-recombination 
processes, but the \lf noise component is absent [42]. However, the component related to \lf noise 
can appear at high temperatures more than 200 K [43]. Such a situation can be explained by the 
fact that the noise behavior of a JFET device having the p-n junction cannot be characterized by the 
semiconductor-oxide surface effects since its channel is separated by the depletion region localized 
along the device channel. 



2.4 MESFETs AND HEMTs 

2.4.1 Small-Signal Equivalent Circuit 

The small-signal equivalent circuit shown in Figure 2.19 proves to be an adequate representation for 
MESFETs and HEMTs in a frequency range up at least to 25 GHz. Here, the extrinsic elements R g , 
L g , Ri, Li, R s , and L s are the bulk and ohmic resistances and lead inductances associated with the 
gate, drain, and source, while C gp and Cj p are the gate and source pad capacitances, respectively. 
The capacitance Cdsd and resistance R^ model the dispersion of the MESFET or HEMT 1-V 
characteristics due to the trapping effect in the device channel, which leads to discrepancies between 
dc measurement and S-parameter measurement at high frequencies [44—46]. The intrinsic model is 
described by the channel charging resistance R gs , which represents the resistive path for charging 
of the gate-source capacitance C gs , the feedback gate-drain capacitance C g d, the output differential 
resistance R As , the drain-source capacitance Cj s and the transconductance g m . The gate-source 
capacitance C gs and gate-drain capacitance C g a represent the charge depletion region and are nonlinear 
functions. The influence of the drain-source capacitance on the device behavior is insignificant 
and its value is practically bias independent. To model the transit time of electrons along the channel, 
the transconductance g m usually includes the time constant r. 




(2.54) 



84 



ACTIVE DEVICES AND MODELING 



S O 




O s 



FIGURE 2.19 Small-signal equivalent circuit of field-effect transistor. 



To describe accurately the small-signal and large-signal device models, it is necessary to satisfy 
charge conservation condition. The models for the device gate-source capacitance C gs and gate-drain 
capacitance C g a should be derived from the charge model. There are commonly four partial derivatives 
of the two device terminal charges, the gate charge Q g and the drain charge Q A with regard to V gs and 
Vds, which appropriately represent totally four capacitances, as shown Figure 2.20(a) [45]. However, 



g O- 



<• <► 



* O d 



dQi ioV^ 



dQJdV* 



o 



g o 



^ O d 




FIGURE 2.20 Capacitance equivalent circuits consistent with charge conservation. 



MESFETs AND HEMTs 85 



the intrinsic small-signal equivalent circuit contains only two capacitances. Consequently, in this 
case, they can be defined as 



9(6g+ 6i) 

dV gs 



Cgd = 



3(G g + Gd) 



dV„ 



(2.55) 



The admittance Y c -matrix for such a capacitive two-port network is 



Yc. = 



ja> (C gs + C gd ) -ywCgd 
_-;'ft)(C gs - C m ) /<wC gd 

where C m is an additional transcapacitance, which is determined by 



r — 



92g 
3V«d 



9g d 
9K S ' 



(2.56) 



(2.57) 



By adding the transcapacitance C m , the capacitance equivalent circuit becomes consistent with the 
charge conservation condition, as shown in Figure 2.20(A). Given that, for the MESFET devices, the 
transcapacitance C m is substantially less than the gate-source capacitance C gs in a frequency range 
where u> < &> T , it can be translated to an additional delay time t c by its combining with the small-signal 
transconductance g m according to g m — jcoC m g m exp(— ja>x c ), where t c = C m /(cojC g!i ). 



2.4.2 Determination of Equivalent Circuit Elements 

To characterize the nonlinear device electrical properties, first we consider the admittance Y- 
parameters derived from the intrinsic small-signal equivalent circuit as 



Yn 
Y u 
Yn 



ja)C g 



1 + ja>C gs R gs 

-j(oC gd 

g m exp(-y<MT) 
1 + ja>C gs R gs 

1 



+ ja>Cgi 



+ ja>C gi 



Y22 = — + jo (C ds + Cgd) 

«ds 



(2.58) 
(2.59) 
(2.60) 

(2.61) 



By dividing these equations into their real and imaginary parts, the parameters of the small-signal 
equivalent circuit can be determined as [47] 



C 2 d = 



ImY 



12 



ImFn — coC g , 



CO 



1 + 



ReJu 



Im Yn — coC. 



g d 



ReFu 

x 2 



(imFn-ojCgo) +(ReF„) 2 
J(ReY 21 ) 2 + (Imy 21 + coC gi ) 2 Jl + {coC gs R g ,) 2 



x — — sin 



-mC,,, 



lmY 2[ - wC gs R gs ReY 2l 



(2.62) 

(2.63) 

(2.64) 

(2.65) 
(2.66) 



86 



ACTIVE DEVICES AND MODELING 



Im Yn 2 — o>C„a 

Cds = 2 1 il (2.67) 

CO 

R, = (2.68) 

Equations (2.62) to (2.68) are valid for the entire frequency range and for the drain voltages of Vi s 
> 0. If we assume that all extrinsic parasitic elements are already known, the only remaining problem 
is to determine the admittance F-parameters of the intrinsic two-port network from experimental data. 
Consecutive stages shown in Figure 2.21 can represent such a determination procedure [48]: 



• Measurement of the S-parameters of the extrinsic device. 

• Transformation of the S-parameters to the impedance Z-parameters with subtraction of the 
series inductances L g and L d . 

• Transformation of the impedance Z-parameters to the admittance F-parameters with subtraction 
of the parallel capacitances C gp and Cd P 

• Transformation of the admittance F-parameters to the impedance Z-parameters with subtraction 
of the series resistances R g , R s , i?d, and inductance L s . 

• Transformation of the impedance Z-parameters to the admittance F-parameters of the intrinsic 
device two-port network. 



The device extrinsic parasitic elements can be directly determined from measurements performed 
at Fds — 0. Figure 2.22 shows the distributed RC channel network under the device gate for zero drain 
bias condition, where AC g is the distributed gate capacitance, Ai?diode is the distributed Schottky 
diode resistance, and AR C is the distributed channel resistance. For any gate bias voltages, by taking 
into account the negligible influence of C gp and Cd P , the extrinsic impedance Z-parameters are 

R c nkT . , 

Z n = R s + R g + + — +j(o(L, + L g ) (2.69) 
3 qlg 

Z l2 = Z 21 = R s + y + jeoL, (2.70) 

Z 22 = R s + R d + R c + jco (L s + L d ) (2.71) 



where R c is the total channel resistance under the gate, nkT/qI g is the differential resistance of the 
Schottky diode, n is the ideality factor, k is the Boltzmann constant, T is the Kelvin temperature, q 
is the electron charge, and / g is the dc gate current. As a result, if the parasitic inductance L s can be 
determined directly from measured ImZn, the parasitic inductances L g and L d are calculated from 
measured ImZn and ImZ22, respectively. The resistance R c is the channel technological parameter, 
which is usually known. The measured real parts of Z-parameters yield the values of R s , R g , and R^. 

At zero drain bias and for the gate voltages lower than the pinch-off voltage V v , the small-signal 
equivalent circuit can be simplified to the one shown in Figure 2.23. Here, the capacitance C\, 
represents the fringing capacitance due to the depleted layer extension at each side of the gate. For 
low-frequency measurements usually up to a few gigahertz, when the extrinsic parasitic resistances 
and inductances have no influence on the device behavior, the imaginary parts of the F-parameters 
can be written as 

ImF n = jco (Cgp + 2C b ) (2.72) 
ImF 12 = ImF 21 = -jcoC b (2.73) 
lmY 22 = jco(C b + C ip ). (2.74) 



MESFETs AND HEMTs 87 




C sp — i— 



s O- 



R, 



—i— Qp 



^11 ^2 



-O s 



g O- 



' sp — i— 



S O- 



Intrinsic 
de\ ice 



(b) 



Intrinsic 
de\ ice 



-O d 



—I— ^ dp 



z 2i Z n -jml^ 



-O s 



s O 





Intrinsic 
device 


( 


) 




) . 


i 



(c) 



-O d 



-O s 



O s 
(d) 



FIGURE 2.21 Method for extracting device intrinsic Z-parameters. 



88 



ACTIVE DEVICES AND MODELING 



Gate 



\ AC,= 








• • • — 






< 


> — < 


> — VWH 


1 — 1 


i — ... — — < 


» — ( 


» Wv 



AR„ 



FIGURE 2.22 Distributed RC channel network schematic under device gate. 

i g R t r h Rj u 




FIGURE 2.23 Small-signal FET circuit at zero drain bias voltage. 



2.4.3 Curtice Quadratic Nonlinear Model 

One of the first simple nonlinear intrinsic large-signal models for a MESFET device for use in the 
design of GaAs integrated circuits is shown in Figure 2.24 [49]. It consists of a voltage-controlled 
source Ifc(V gs , Vd S ), the gate-source capacitance C gs (V gs ), and a clamping diode between gate and 
source. The gate-drain capacitance C g d is assumed to be constant. 

An analytical function proposed to describe the nonlinear current source behavior is 



/da = P {V gs - V p ) (1+1 V is ) tanh (of V ds ) 



(2.75) 



8 O 




O d 



FIGURE 2.24 Curtice quadratic nonlinear intrinsic model. 



MESFETs AND HEMTs 89 



where /S is the transconductance parameter determined from experimental data, V p is the pinch-off 
voltage, and X is the slope of the drain characteristic in the saturated region. Due to the finite value 
of maximum charge velocity of about 10 7 cm/s during transient operation, change in the gate voltage 
does not cause an instantaneous change in the drain current. For example, it takes the order of 10 
ps to change the drain current after the gate voltage is changed in a 1-p.m gate-length MESFET. 
Consequently, the most important result of this effect is a time delay between gate-source voltage 
and drain current. Therefore, the current source given by Eq. (2.75) as 7ds[V gs (f), VdsW] should be 
altered to be 7ds[Vgs(f — t), VdsML where r is equal to the transit time under the gate. To more 
accurate approximate the drain current behavior, a cubic nonlinear model can be used [50]. 

The time delay effect is not easily added to most circuit analysis program. Therefore, a simple 
and sufficiently accurate way to solve this problem is to assume the current source to be of the form 



where the second term is considered the first term of the Taylor series expansion of / ds (? — r) in time 
when for small r the error is quite small, 



The gate-source capacitance C gs and gate-drain capacitance C g d can be treated the voltage- 
dependent Schottky-barrier diode capacitances. For the negative gate-source and small drain-source 
voltages, these capacitances are practically equal. However, when the drain-source voltage is in- 
creased beyond the current saturation point, the gate-drain capacitance C g d is much more heavily 
back-biased than the gate-source capacitance C gs . Therefore, the gate-source capacitance C gs is sig- 
nificantly more important and usually dominates the input impedance of the MESFET device. In this 
case, an analytical expression to approximate the gate-source capacitance C gs is 



where C gs o is the gate-source capacitance for V gs = 0 and V gsi is the built-in gate voltage. When V gs 
approaches V gS i, the denominator in Eq. (2.77) must not be allowed to approach zero since C gs will 
continue to increase as the depletion width reduces, so that a forward bias condition occurs, and the 
diffusion gate-source capacitance becomes of great importance. The built-in voltage V gsi should be 
equal to the built-in voltage of the Schottky-barrier junction plus some part of the voltage drop along 
the channel under the gate. 

2.4.4 Parker-Skellern Nonlinear Model 

The comprehensive large-signal Parker-Skellern model, which intrinsic nonlinear part is shown in 
Figure 2.25, is a realistic description of measured characteristics of the MESFET device over all 
its operation regions [51]. The model maintains strict continuity in high-order derivatives, describes 
subthreshold and breakdown regions, self-heating effect, and frequency dependence of the output 
conductance and transconductance. The model parameters provide independent fitting to all operation 
regions. Both the gate-source capacitance C gs and the gate-drain capacitance C g d are treated as the 
standard voltage-dependent depletion capacitances with sensitivity y — 0.5. 



(Vg.) 



dlj. (Vg) 

T dt 



(2.76) 



(Vg,) = dhs (V) dV, 
dt ~ dV gs % dt 




(2.77) 



90 



ACTIVE DEVICES AND MODELING 



B O 




O d 



FIGURE 2.25 Parker-Skellern nonlinear intrinsic model. 



An analytical function to describe the nonlinear current source behavior is proposed as 

, Q" 



lis = 



EL 

1 + SX 



1 



(2.78) 



where 



V s t) 



dX 



(2.79) 



with 



K T = Kx(l+MV ds )ln 



1 + exp 



V gs - V, h - Yi Vgd - yi (V gd - VV) - y 3 ( Vg S - V gl 



dV gs — 

T g V gd = V gi - T g 



dVp 
dt 



v dP = V is 



v, 



2 V?>- 

? (*> - V,,,) VgT 



P-Q 



where fi is the linear-region transconductance scaling coefficient, S is the thermal reduction coefficient, 
r d is the relaxation time for thermal reduction, r g is the relaxation time for gamma feedback, M is 
the subthreshold modulation, P is the linear-region power-law exponent, Q is the saturated-region 
power-law exponent, V& is the threshold voltage, V s t is the subthreshold potential, Z is the knee 



MESFETs AND HEMTs 



91 



transition parameter, <p is the gate-junction potential, £ is the saturation-knee potential factor, and y\, 
y 2 , and y 3 are the fitting parameters. 

The model must select V sa t to be less than V e j and the velocity saturation potential that is set 
by the parameter £ and the channel depletion potential <p — For a 1-iim GaAs MESFET, £ is 
approximately 0.3. A single set of dual power-law model parameters P and Q can describe both the 
controlled-resistance and controlled-current regions. The drain conductance in a controlled-current 
region is fitted with parameter P, whereas the transconductance in a controlled-current region is 
fitted with parameter Q. To describe the frequency dispersion effects producing differences between 
the drain conductance at dc and at high frequencies, the model determines the bias condition by a 
continuous calculation of the average value of the thermal potentials K gs and V g(1 , where the average 
is accumulated over a user-defined time constant r g which is normally between 0.1 to 1 ms. The 
average potential at time t is calculated by a weighted sum of the instantaneous potential at time t 
and previously calculated average at time (t — At) as 



where V(t) is replaced by V gs (i) or V g i(t) and r is replaced by r g or rj. At very low frequencies, the 
instantaneous and average potentials are identical. 

The gate junction current is implemented with identical gate-source and gate-drain diodes between 
the intrinsic nodes. The term S is a model parameter related to the product of the temperature coefficient 
of current reduction and the channel resistance. The model calculates average dissipated power rather 
than using instantaneous power over a time constant r d using algorithm of Eq. (2.80). 

2.4.5 Chalmers (Angelov) Nonlinear Model 

A simple and accurate large-signal model for different submicrometer gate-length HEMT devices 
and commercially available MESFETs, which is capable of modeling the drain current-voltage char- 
acteristics and its derivatives, as well as the gate-source and gate-drain capacitances, is shown in 
Figure 2.26 [52,53]. This model can be used not only for large-signal analysis of power amplifier and 
oscillators, but also for predicting the performance of multipliers and mixers including intermodula- 
tion simulation. 




(2.80) 




6 



s 



FIGURE 2.26 Angelov nonlinear intrinsic model. 



92 



ACTIVE DEVICES AND MODELING 



The drain current source is described by using the hyperbolic functions as 

/* = V(l +tanhVO(l + Xl/ ds )tanh(aV ds ) (2.81) 

where / pk is the drain current at maximum transconductance with the contribution from the output 
conductance subtracted, X is the channel length modulation parameter, and a — a 0 + aitanhi/f is the 
saturation voltage parameter, where a 0 is the saturation voltage parameter at pinch-off and cxi is the 
saturation voltage parameter at V gs > 0. 

The parameter \jr is a power series function centered at Vp k with the bias voltage V gs as a variable, 

f = Pi (V gs - V pk ) + P 2 (V gs - V pk f + P 3 (V gs - V pk ) 3 + ■ • • (2.82) 

where V pk is the gate voltage for maximum transconductance g mp k- The model parameters as a first 
approximation can be easily obtained from the experimental Ifc(V gs , V^) dependencies at a saturated 
channel condition when all higher terms in \jr are assumed to be zero, and X is the slope of the 7 ds -F ds 
curves. 

The intrinsic maximum transconductance g mpk is calculated from the measured maximum 
transconductance g mpkm , by taking into account feedback effect due to the source resistance R s , 
as 

gmpk = gm ^ (2.83) 

1 — gmpkm-"s 

To evaluate the gate voltage V pk and parameter Pi, it is necessary to define the derivatives of the 
drain current. If higher order terms of \jr are neglected, the transconductance g m becomes equal to 

gm = w~ = IpkP[ sech [ Pl ( Vgs ~ Vpk ^ 2 (1 + XVis) tanh (al/ds) • (2,84) 

The gate voltage V pk that depends on the drain voltage can be extracted by finding the gate voltages 
for maximum transconductance, at which the second derivative of the drain current is equal to zero. 
In this case, it is advisable to use the simplified expression 

V pk (V ds ) = V pk0 + (V pks - V pi0 ) (1 + AV ds )tanh(aV ds ) (2.85) 

where F pk o is measured at V ds closely to zero and V pks is measured at V ds in the saturation region. 
A good fitting of P\ and good results in harmonic balance simulations are obtained using 



P\ = Pi, 



1 + 



Pio -\ 1 



Asa, /C0Sh 2 (BV ds ) 



(2.86) 



where Pio = gmo//pko at Va s close to zero and B is the fitting parameter (B 1.5a). The parameter 
P2 makes the derivative of the drain current asymmetric, whereas the parameter P3 changes the drain 
current values at voltages V gs close to pinch-off voltage V p . Three terms in Eq. (2.82) are usually 
enough to describe the behavior of the different MESFET or HEMT devices with acceptable accuracy. 

The same hyperbolic functions can be used to model the intrinsic device capacitances. When 
5-10% accuracy is sufficient, the gate-source capacitance C gs and gate-drain capacitance C gd can be 
described by 

C gs = C gs0 [1 + tanh (Pi gsg V gs )] [1 + tanh (P lgsd V ds )] (2.87) 
C gd = C gd0 [1 + tanh (Pig dg V gs )] [1 - tanh (P lgdd V ds + P icc V gs V As )] (2.88) 



MESFETs AND HEMTs 93 




o i . 1 1 . 

-10 12 3 I*, V 



FIGURE 2.27 Gate-source and gate-drain capacitances versus drain-source voltage. 



where the product Pi C cV gs Vi S reflects the cross-coupling of V gs and Vd S on C g d and the coefficients 
^lgsg, ^tgsd, f'igdg, and Pigdd are the fitting parameters. These dependencies, unlike the commonly 
diode-like models, are suitable for HEMT devices with an undoped AlGaAs spacer-layer in view of 
the saturation effect of the gate-source capacitance C gs for increasing V gs . This is due to the absence 
of parasitic MESFET channel formation in the undoped AlGaAs layer, found in HEMTs with a doped 
AlGaAs layer. The approximate behavior of normalized gate-source capacitance C gs /C gs o (curve 1) 
and gate-drain capacitance C g d/C g do (curve 2) as functions of V^ds for zero gate-source voltage is 
shown in Figure 2.27, where C gs o and C g do are the gate-source and gate-drain capacitances at V d s = 
0, respectively. The character of the curves is the same for both positive and negative V gs , except that 
the capacitance range decreases for the same range of Vds with the decrease of V gs . 

The drain-source dispersive resistance R As<i as a nonlinear function of the gate-source voltage V gs 
can be defined by 

7?dsdp 

^dsd = ^dsdO + — — (2.89) 

1 + tanht/f 

where 7?dsdo is the minimum value of 7? d sd and 7?dsdp determines the value of R AsA at the pinch-off [46]. 
2.4.6 IAF (Berroth) Nonlinear Model 

An analytical charge conservative large signal model for HEMT devices, which is valid for a frequency 
range up to 60 GHz, is shown in Figure 2.28 [54]. In this model, the current sources / gs , 7 g d, and 7d s 
and capacitances C gs and C g d are considered as the nonlinear elements. 

The drain current source is represented by the following nonlinear equation with 10 fitting param- 
eters: 



/ds = / (Vgs) 



1 + -, rV is 

1 + A A {V gs -V c + 2/p) 



tanh(aV ds ) (2.90) 



where 

/ (V„) = CD VC { 1 + tanh [/J (V gs - V c ) + y (V gs - V c ) 3] } + CD mb { 1 + tanh [S (V gs - V sb )] } 



a is the slope of drain current in the pinch-off region, f} is the slope parameter of drain current, y 
is the slope parameter of drain current in the pinch-off region, X is the slope of drain current in the 



94 



ACTIVE DEVICES AND MODELING 



S O 




o d 



6 

s 



FIGURE 2.28 Berroth nonlinear intrinsic model. 



saturation region, A A is the gate voltage parameter for slope of drain current, & is the drain current 
slope parameter correction term, V c is the gate voltage for maximum transconductance, Va, is the 
gate voltage for maximum transconductance correction term, CD VC is the drain current multiplication 
factor, and CZ> VS b is the drain current multiplication factor correction term. 

To describe the / gs and 7 g d current sources, the diode model for both forward and reverse bias 
operation modes was used in the form 



where 7d sat is the forward bias fitting parameter, Vt is the temperature voltage, and n is the diode 
ideality factor. 

The nonlinear capacitances C gs and C g d are calculated by differentiating the voltage-depending 
charge function 2 g (V gs , Va s ) with respect to V gs and Vd S , which leads to the input capacitance C\\ — 
C gs + C g d and transcapacitance Cn — — C g d, respectively, 



f 2 = l + Dlncosh(FV ds ) 
V\ is the transition voltage, and A, B, C, D, E, and F are the model-fitting parameters. 

2.4.7 Noise Model 

The noise properties of a MESFET device can be described based on both its physical and equivalent 
circuit models. The dominant intrinsic noise of a microwave GaAs MESFET device is the diffusion 
noise introduced by electrons experiencing velocity saturation. In a device two-zone model, a portion 
of the channel near the source end is assumed to be in the constant mobility operation mode (zone I), 
while the remaining portion near the drain end is postulated to be in velocity saturation (zone II). The 
position of the boundary between these zones is a strong function of the source-drain bias with weak 
dependence on the gate-source bias. It is assumed that the noise in zone I is thermal enhanced by hot 
electron effects [55,56]. However, zone II cannot be treated as an ohmic conductor. Its contribution 
must be represented as a high-field diffusion noise, being dominant in microwave devices [57]. This 
diffusion noise is proportional to the high-field diffusion coefficient and is linearly dependent on drain 
current. On the other hand, the thermal noise of zone I decreases with increasing drain current. As a 
result, a strong correlation exists between the drain noise and the induced gate noise which leads to 
a high degree of cancellation in the noise output of the GaAs MESFET [58]. 




(2.91) 



Gg (V gs ,V ds ) = AfJ 2 + E (V gs - 0.5 V ds ) 



(2.92) 



where 



/j = ^lncosh{S [(7 gs - Vi) - 0.5tanh(Cl/ ds )]} + (V gs - Vi) - 0.5 tanh (CV is ) 



MESFETs AND HEMTs 95 




FIGURE 2.29 MESFET equivalent circuits with noise sources. 



The noise equivalent circuit of the MESFET device with both intrinsic and extrinsic noise sources 
is shown in Figure 2.29(a) [55,58]. The noise source ijL represents the noise induced on the gate 
electrode by the passing thermal fluctuations in the drain current. The intrinsic drain noise source 
/„ d has a flat spectrum. The resistance R gs represents the resistive charging path for the gate-source 
capacitance C gs , and noise associated with this resistor is imbedded in the gate noise source. The 
series gate, source and drain resistances are represented by the voltage thermal noise sources e^ g , e^, 
and e\ A , respectively. 

The noise voltage and current sources can be given through their mean-square values as 



AkTR g Af 
4kT(coC gs ) 2 RAf 



(2.93) 
(2.94) 



A. - 



4kTR s Af 
AkTRiAf 
4kTg m PAf 



(2.95) 
(2.96) 
(2.97) 



96 



ACTIVE DEVICES AND MODELING 



where R and P are the gate and drain noise model parameters depending upon the implementation 
technology and biasing conditions [59,60]. 

The cross-correlation between the gate-drain noise current sources i% and i^, can be written as 



f'ndi'ng = 4kTcoC m CVPRAf 



(2.98) 



where C is the correlation coefficient. The quantities P, R, and C are the bias-dependent empirical 
correction factors, which may be obtained by noise de-embedding techniques [61]. Their typical 
values based on measurement and calculation of the noise figure for different devices can be chosen 
as P — 1, R — 0.5, and C — 0.9 [62]. It should be noted that and C increase in ohmic region 
and tend to saturate at high drain voltage, while i\ increases with a near constant slope versus drain 
voltage. 

The noise current source i\ gX is responsible for the effect of the gate leakage current, which should 
be taken into account when using a submicrometer gate-length HEMT device. It can be written as a 
shot noise source in the form 



'ngi = 2a?/giA/ 



(2.99) 



where / gl is the gate leakage current and a is the fitting parameter [63,64]. 

The admittance ^-parameters of the MESFET intrinsic small-signal equivalent circuit can be 
written in matrix form 



jwC g 



1 + j(oR gs C gs 
g m exp(-;&)r) 
1 + ./'n)tfg S C gs 



-jcoC, 



+ j<» (C ds + C gd ) 



(2.100) 



The corresponding admittance noise correlation matrix to calculate the equivalent noise resistance 
R n , optimal source admittance Fsopt. and minimum noise figure F min is given by 



C Y = 4 AT 



a hi (a>Ces) R , — 
" jcoC gs CVPR 
2V T g m 



-jcoC gs CVPR 



gmP 



(2.101) 



where Vt = kT/q is the thermal voltage. However, if the correlation matrix has been determined, 
the noise parameters can be analytically calculated by using the elements of the chain correlation 
matrix [65] 



r A - 



^19 — 



L 9 



\y 2 i r 



^99 — 



<- I 



\Yn\ 2 C] 2 



\Y 2 i\ 



1 1 1 *- 21 _ 1 11 L 12 
Y2I ^9*1 



(2.102) 
(2.103) 
(2.104) 



In a first approximation, the gate noise source ;'^ g , feedback capacitance C g d, and series drain re- 
sistance Ri can be neglected. As a result, the simple approximate expressions based on measurements 



BJTs AND HBTs 97 



can be obtained in terms of the device equivalent circuit elements as 



1 + 0.016 fC 




R g + R, 



(2.105) 



0.8 



R 



(2.106) 



gm 




+ R g + R s 



) 



(2.107) 



160 



(2.108) 



provided that R n , Rs opl , X Sopt , R g , and i? s are in ohms, transconductance g m in mhos, capacitance C gs 
in picofarads, and operating frequency/ is in gigahertz [66,67], 

The minimum noise figure given by Eq. (2. 105) can also be expressed in terms of the device 
geometrical parameters as 



where the effective gate length L is in micrometers [68]. 

A comparison of the noise performance of both HEMT and conventional MESFET devices 
demonstrates the HEMT superiority, mainly related to its higher transition frequency and correlation 
coefficient [62]. The transition frequency of HEMT device is greater for two main reasons: higher 
carrier mobility results in a higher average velocity and, therefore, a higher transconductance, whereas 
the small epilayer thickness yields higher transconductance and less effect of the parasitic capaci- 
tances. In addition, the correlation coefficient C is close to 0.7-0.8 for short-gate-length MESFETs, 
but becomes close to 0.8-0.95 for HEMTs. 

The GaAs MESFET devices are characterized by a significant value of the flicker noise. This is a 
combined result of the gate leakage current, fluctuations of the Schottky barrier space charge region, 
and carrier number fluctuations in the channel and at the interface between the channel and substrate 
due to trapping phenomena [69]. The trapping mechanism is especially pronounced in GaAs material 
where the trapping centers can arise from a variety of causes such as trace impurities and crystal 
defects [70]. It is shown that the low-frequency noise in GaAs MESFET devices on InP substrate 
is directly related to the structural quality of GaAs active layers when increasing the buffer layer 
thickness for GaAs lattice mismatched on InP substrate improves the noise performance [71]. 

2.5 BJTs AND HBTs 

2.5.1 Small-Signal Equivalent Circuit 

The complete bipolar transistor small-signal equivalent circuit with extrinsic parasitic elements is 
shown in Figure 2.30. Based on this hybrid 7r-type representation, the electrical properties of the 
bipolar transistors, in particularly HBT devices, can be described with sufficient accuracy up to 
30 GHz [72,73]. Here, the extrinsic elements R b , Lb, R c , L c , R e , and L e are the series resistances and 
lead inductances associated with the base, collector, and emitter; and C p b e , C p bc, and C pce are the 
parasitic capacitances associated with the contact pads, respectively. The lateral resistance with the 
base semiconductor resistance underneath the base contact and the base semiconductor resistance 
underneath the emitter are combined into a base-spreading resistance r b . The intrinsic model is 
described by the dynamic diode resistance r n , the total base-emitter junction capacitance and base 
charging capacitance C n , the transconductance g m , and the output Early resistance r ce that model the 




(2.109) 



98 



ACTIVE DEVICES AND MODELING 



(■■"phi 



1 t-pllL 



''l-i 



Intrinsic device 



e O- 



-O e 



FIGURE 2.30 Small-signal equivalent circuit of bipolar device. 



effect of base- width modulation on the transistor characteristics due to variations in the collector-base 
depletion region. To increase the usable operating frequency range of the device up to 50 GHz, it is 
necessary to properly include the collector-current delay time in the current source as g m exp(— jcDT„), 
where is the transit time [74]. 

2.5.2 Determination of Equivalent Circuit Elements 

If all extrinsic parasitic elements of the device equivalent circuit shown in Figure 2.30 are known, 
the intrinsic two-port network parameters can be embedded with the following determination proce- 
dure [72]: 

• Measurement of the S-parameters of the transistor with extrinsic elements. 

• Transformation of the S-parameters to the admittance F-parameters with subtraction of the 
parasitic shunt capacitances C p b e , C P b C , and C pce . 

• Transformation of the new F-parameters to the impedance Z-parameters with subtraction of the 
parasitic series elements L b , R b , L e , i? e , L c , and R c . 

• Transformation of the new Z-parameters to the F-parameters with subtraction of the parasitic 
shunt capacitance C co . 

• Transformation of the new F-parameters to the Z-parameters with subtraction of the parasitic 
series resistance /v 

• Transformation of the new Z-parameters to the F-parameters of the intrinsic device two-port 
network. 

The bipolar transistor intrinsic F-parameters can be written as 

Fn = — + jo)(C w +C ci ) (2.110) 

Y u = -ja>C d (2.111) 
F 2 i = gmexpt-jftri^) + ja)C cl (2.112) 

Yn=—+jcoC A . (2.113) 

''ce 



BJTs AND HBTs 99 



After separating Eqs. (2.110) to (2.113) into their real and imaginary parts, the elements of the 
intrinsic small-signal equivalent circuit can be determined analytically as 



lm(Y n + Y l2 ) 



CO 

1 



ReF 



(2.114) 
(2.115) 



ii 



ImFp 

C ci = (2.116) 



(ReF 21 ) 2 + (ImF 21 + ImF 12 ) 2 (2.117) 



1 ReF 21 + ReYv 



cos' 1 (2.118) 

(2.119) 



x/(ReF 21 ) 2 + (ImF 21 + ImF 12 ) 2 
1 



ReF 22 

The parasitic capacitances associated with the pads can be determined by the measurement of the 
open test structure with the corresponding circuit model shown in Figure 2.31(a). When the values of 
these pad capacitances are known, it is easy to determine the values of the parasitic series inductances 
by measuring the shorted test structure with corresponding circuit model shown in Figure 2.31(b). 

The values of series parasitic resistances can be calculated on the basis of physical parameters of 
the device or by adding them to the intrinsic device parameters (with the appropriate solution of a 
nonlinear system of eight equations with eight independent variables using iterative technique) [73]. 
In the latter case, it is assumed that influence of the transit time r„ on the bipolar electrical properties 
in a frequency range up to 30 GHz is negligible. 

The external parasitic parallel capacitance C co , as well as the other device capacitances, can be 
estimated from the device behavior at low frequencies and cutoff operating conditions [75,76]. For 



b O 



e O- 




O c 



* O e 



c O- 



— O e 



FIGURE 2.31 Models for parasitic (a) pad capacitances and (b) lead inductances. 



100 



ACTIVE DEVICES AND MODELING 



C',,b, 



b o- 



-O c 



r he -|— I t-'jl —I— Cp,-; 



c O- 



-O c 



FIGURE 2.32 Small-signal model at low frequencies and cutoff operation mode. 



such conditions, the device small-signal equivalent circuit is reduced to capacitive elements, as shown 
in Figure 2.32. 

The device capacitances can be directly calculated from measured 7-parameters by 



Cpbe "T" C n — 



C 



Cpbc "1" Ceo "1" Cci 



Im(y„+F 12 ) 



Im(7 u +y 12 ) 

CO 

ImF 12 

CO 



(2.120) 
(2.121) 
(2.122) 



Since C pDC and C c0 are the bias-independent capacitances and C c ; is the base-collector junction 
capacitance, the extraction of C P b C + C co can be carried out by fitting the sum C P b C + C co + C C i 
to the expression for junction capacitance at different base-collector voltages. If an approximation 
expression for C co is given by 



Cco — Cj C o/. / 1 "I" 



Vbc 
(Pc 



(2.123) 



then the extraction of the parameters Cj C0 , <p c , and C C j can be performed using the linear equation 

2 

1 

(2.124) 



ImYv 



H~ Cpbc H~ Cci 



1+ - V bc . 

Vc 



As a result, linearizing this equation by choosing a proper value for Cd with known value of C P b C 
gives the values for remaining two parameters C co and cp c from the slope and intercept point of the 
final linearized dependence. 



2.5.3 Equivalence of Intrinsic n- and T-Type Topologies 

The small-signal equivalent circuit of a bipolar transistor can be represented by both 7r-type and 
J-type topologies. The T-type equivalent circuit representation is appealing because all the model 
parameters can be directly tied to the physics of the device and an excellent fit between measured 
and simulated S-parameters in the frequency range up to 30-40 GHz [74-77], The HBT small-signal 
equivalent circuit with 7-type topology is shown in Figure 2.33. 



BJTs AND HBTs 101 




FIGURE 2.33 Small-signal T-model of bipolar device. 



There is one-to-one correspondence between 7r-type and T-type device models. By comparing 
both small-signal equivalent circuits shown in Figures 2.30 and 2.33, the difference, which can be 
found in the representation of both intrinsic device models, are just enclosed in dashed boxes. From 
Figure 2.34 it follows that the admittances Y e — IJV\,t for 71- and T-type models are defined by 

1 1 

Y e = — + ja>C e = h jcoC n + g m exp (-jcax„) . (2.125) 

'"e 'V 

The collector source currents for both models are the same, 



<* exp (-jcor tee ) / e = g m exp (- /cut*) V n (2.126) 

where r tee is the transit time for a T-type model, a — a 0 /(l + j(OT a ), x a — \l(2nf a ),f a is the alpha 
cutoff frequency, and c*o is the low frequency collector-to-emitter current gain. 



b O 



c O- 



-O I! 



-Oc 



bO 



<^> f'he < 'c —i— ( t 



(a) 



<a — oc 



6 



a cxp(-/wt, c0 V c 



FIGURE 2.34 Intrinsic (a) jr-type and (h) T-type device topologies. 



102 



ACTIVE DEVICES AND MODELING 



The expressions for intrinsic n -model parameters can be derived through the intrinsic T-model 
parameters as [74] 



a 0 



e>m i T i I 

Vl + (f0T a ) 2 V V. 
1 

Tjy — Ttee 

CO 

1 1 

— = gm COS (0)0 

'V r e 



[tan 1 (aiC e r e ) + tan 1 (a>r a )] 



C — C 



sin (cor n ) 



(2.127) 

(2.128) 
(2.129) 
(2.130) 



Both 7r-type and T-type bipolar device topologies can describe the transistor electrical properties 
in a very wide frequency range and, when optimized, up to 50 GHz. 



2.5.4 Nonlinear Bipolar Device Modeling 

Since the bipolar transistor can be considered to be an interacting pair of p-n junctions, the approach 
to model its nonlinear properties is the same as that used for the diode modeling. The simple large- 
signal Ebers-Moll model with a single current source between the collector and the emitter is shown 
in Figure 2.35 [1]. The collector-emitter source current / ce is defined by 



exp 



exp 



(2.131) 



where / sat is the bipolar transistor saturation current and Vj is the thermal voltage calculated from 
Eq. (2.2). 

The device terminal currents are defined as / c — I ce - I bc , I e — -I ce - / be , and / b = / be + I be > where 
the diode currents are given by 



he = 



^sat 



exp 



(2.132) 
(2.133) 



h Kb 
b O * 




FIGURE 2.35 Large-signal Ebers-Moll model. 



BJTs AND HBTs 103 



where /S F and /3 R are the large-signal forward current gain and reverse current gain of a common 
emitter bipolar transistor, respectively. 

The device capacitances C x and Cbc each consist of two components and are modeled by the 
diffusion capacitance and junction capacitance, respectively, as 



where tp and tr are the ideal total forward time and reverse transit time, Cj eo and Cj C0 are the 
base-emitter and base-collector zero-bias junction capacitances, and m e and m c are the base-emitter 
and base-collector junction grading factors, respectively. 

The collector-emitter substrate capacitance C ce should be taken into account when designing 
integrated circuits. Its representation is adequate for many cases, since the epitaxial-layer-substrate 
junction is reversed-biased for isolation purposes, and usually it is modeled as a capacitance with 
constant value. 

The Ebers-Moll model cannot describe the second-order effects due to low current and high-level 
injection, such as a base-width modulation (Early effect) and variation of the large-signal forward 
current gain /3 F with collector current / ce . In addition, to find a better approximation of the distributed 
structure of the base-collector junction at microwave frequencies, the junction capacitance Cb C 
should be divided into two separate capacitances: internal C C i and external C co . The lateral resistance 
and the base semiconductor resistance underneath the base contact and the base semiconductor 
resistance underneath the emitter are combined into a base-spreading resistance r b . Figure 2.36 
shows the modified 7r-type Gummel-Poon nonlinear model of the bipolar transistor that can describe 
the nonlinear electrical behavior of bipolar transistors, in particularly HBT devices, with sufficient 
accuracy up to 20 GHz [78,79]. 

For the Gummel-Poon large-signal model, the collector source current / ce is determined by 




(2.134) 




(2.135) 




(2.136) 




b O — WV 



+ 



A/Vv — O c 



v-.y. =^c. 



R, 



■e 




a 

FIGURE 2.36 Large-signal Gummel-Poon model. 



104 



ACTIVE DEVICES AND MODELING 



where 7 SS is the BJT fundamental constant defined at zero-bias condition, « F is the forward current 
emission coefficient, n R is the reverse current emission coefficient, and q b is the variable parameter 
defined by 



where 



<7i 



(2.137) 



Ikf 



exp 



1 



/ ss 




+ f- 

'KR 


exp ^ 





1 



V A is the forward Early voltage, V B is the reverse Early voltage, /kf is the knee current for high-level 
injection in the normal active region, and /kr is the knee current for low-level injection in inverse 
region [1]. 

The currents through model diodes are defined by 



Ihe. — 



/sat (0) 

/6fm(0) L 

/sa.(0) 



exp 



(—) 



1 



[(0) 



exp 



n R V T 



+ C 2 / sat (0) 

+ C 4 /sat(0) 



exp 



exp 



«elV t 
«clV t 



1 



(2.138) 
(2.139) 



where / sa t(0) is the saturation current for Vt, c — 0, /2fm(0) and /Jrm(0) are the large-signal forward and 
reverse current gains of a common-emitter BJT in mid-current region for V bc — 0, C2 and C4 are the 
forward and reverse low-current nonideal base current coefficients, respectively, «el is the nonideal 
low-current base-emitter emission coefficient, and « C l is the nonideal low-current base-collector 
emission coefficient. 

The nonlinear behavior of the intrinsic base resistance ; _ b can be described by 



'"b — 'bm + 3 (r t 



r bm) ■ 



tan; 



z tan z z 



(2.140) 



where 



144 7 b 

7T- / rb 




r bm is the minimum base resistance that occurs at high-current level, r h0 is the base resistance at zero 
bias with small base current level, and 7 r b is the current where the base resistance falls halfway to its 
minimum value [1]. 

The intrinsic device capacitances C„, C c \, and C co are modeled by the diffusion capacitance and 
junction capacitance, respectively, as 



r d ( /« 
dVy, \ q b 



+ Cjeo 1 



Cd = T R — — h KcCjco ( 1 

dVbc 



Vbc 



r 

^ LI 



CjcoCl-*,) 1 



Vbco 



(2.141) 
(2.142) 
(2.143) 



BJTs AND HBTs 



105 



where k c is the fraction of the base-collector junction capacitance connected to the base resistance 
>"b, ^bco is the voltage through the capacitance C co , and r FF is the modulated transit time defined by 



where X rF is the transit time bias dependence coefficient, / rF is the high-current parameter for effect 
on r F , V rF is the value of V bc where the exponential equals to 0.5, and 



As it follows from Eq. (2.141), the nonlinear behavior of capacitance C n strongly depends on 
the effect of transit time modulation characterized by t ff . This transit charge variation results in 
significant changes of the transition frequency / T at various operation conditions. For example, at 
medium currents, / T reaches its peak value and is practically constant. Here, the ideal transit time is 
defined by t f = l/(27r/ T ) and the dominated base-emitter diffusion capacitance increases linearly 
with collector current. At low currents, fj is dominated by the junction capacitance and increases 
with the increase in collector current. At high currents, the widening of the charge-neutral base region 
and pushing of the entire space-charge region toward the heavily doped collector region (Kirk effect) 
degrades the frequency response of the transistor by increasing the transit time and decreasing/x. In 
this case, the transit time is modeled by Tpp. 

The more complicated models, such as VBIC, HICUM, or MEXTRAM, include the effects of 
self-heating of a bipolar transistor, take into account the parasitic p-n-p transistor formed by the 
base, collector, and substrate regions, provide an improved description of depletion capacitances at 
large-forward bias, take into account avalanche and tunneling currents, and other nonlinear effects 
corresponding to distributed high-frequency effects [80]. 

2.5.5 Noise Model 

The noise in a bipolar transistor is assumed to arise from three basic sources: diffusion fluctuations, 
recombination fluctuations in the base region, and thermal noise in the base resistance [81]. The noise 
behavior of the bipolar transistor can be described based on its equivalent circuit representation shown 
in Figure 2.37(a), which includes the main elements responsible for the device electrical behavior 
and noise sources. Since the process of the carrier drifting into the collector-base depletion region is 
a random process, the collector current / c demonstrates shot noise and is represented by a shot-noise 
collector current source i^ c . The base current 7 b is a result of the carrier injection from the base 
to the emitter and generation-recombination effect in the base and base-emitter depletion regions. 
Because all these components are independent, representing a random process, the base current also 
demonstrates a shot-noise behavior and is represented by a shot-noise base current source i^ b . Flicker 
noise is represented by a current source across the internal base-emitter junction combined with 
the base current source i* b . The series base, emitter and collector resistances are represented by the 
voltage and current thermal noise sources e\ h , /^ e , and e\ z , respectively. 

The noise voltage and current sources can be given through their mean-square values as 




(2.144) 




(2.145) 



nb 



4kTr b Af 
AkTAf 



(2.146) 



(2.147) 



2qIAf 



(2.148) 



4kTr c Af 



(2.149) 



2qI b Af + K v '-^Af 



(2.150) 



106 



ACTIVE DEVICES AND MODELING 



c 




FIGURE 2.37 Bipolar equivalent circuits with noise sources. 



where q is the electron charge, A F is the flicker noise exponent, K F is the flicker noise coefficient 
calculated as 2c//' c , and/ c is the flicker noise corner frequency [1,82]. 

The required minimum noise figure F^, noise resistance R n , and optimum source admittance 
F Sopt using the noise correlation Ca -matrix parameters as functions of the input-referred noise voltage 

and noise current i^, as well as 7-parameters of the simplified noise-free two-port network shown 
in Figure 2.37(A), can be calculated for a sufficiently high value of the low-frequency current gain 
P = g m r„ from 



R a = r b [l + 



2r„ 



f_ 

20 ■ 2fir x V/t 



(2.151) 



'Sopt 



1 + 



1 ( f 



1 ./' 



J: 



1 / 



2pR n f T J J 2/3R a f T 



(2.152) 



1 + 



1 + 



1 ( f 



fi V/t 



+ 



2r b 



2/v / Br„ 



1 + 



rj (/t 



(2.153) 



where/ is the operation frequency and/ T = g m /(27tC w ) is the bipolar transition frequency (the effect 
of the feedback collector capacitance C c is not taken into account) [83], It should be noted that if the 
optimum noise conductance Gs 0 pt is insensitive to the collector capacitance C c , then it can severely 



REFERENCES 107 



affect the other noise parameters at microwave frequencies, mostly due to the reduction of the device 
gain capability. 

A noise model for HBT device operated at very high frequencies should include the contribution 
of both space-charge layers (at the emitter-base junction and the base-collector junction) to the shot 
noise. These two noise sources related to the collector current / c are the result of the same electrons, 
which are injected from the emitter into the base, cross this layer, and then reach the collector. 
Therefore, their correlation can be given by a time delay function exp(— jcor), where r is the transit 
time through the base and the collector-base junction which is x n for a jr-type model [84-86]. Thus, 
to extend the HBT noise model valid up to its transition frequency, the base and collector noise current 
sources are rewritten as 

& = 2q (/„ + |1 - exp(->r)| 2 /„) Af (2.154) 
i[ c = 2qI c Af (2.155) 
J~J^ C = 2q [exp(-;wt) - l] I c Af. (2.156) 

Flicker noise in bipolar transistors is associated mainly with generation-recombination centers, 
which contribute to random trapping and release of free carriers [87,88]. This relatively slow process 
is always associated with flowing current by which mean-square value ijj as a function of the offset 
frequency / can be approximated 

Ji = K F I k ^- (2.157) 

where K F is the flicker noise coefficient and k is the flicker noise exponent. Both these coefficients 
are device dependent. In conventional high-quality silicon bipolar devices, the low-frequency noise 
is determined by 1// noise in the base current due to carriers injected from the base into the emitter, 
since the emitter series resistance can be neglected, the base series resistance is low and the collector 
current has an ideality factor of 1. In downscaled polysilicon bipolar transistors with lower emitter 
area, at low bias currents the \lf noise in the base current is dominant, while at higher bias currents the 
influence of the series resistances on the noise becomes noticeable. In GaAs/AlGaAs HBT devices, 
the 1/f noise in the base current can be described by Eq. (2.157) with Kp fx 10~ 10 A 2 ~ k and k fx 1.6, 
whereas, for the 1//' noise in the collector current, K F fx 10~ 12 A 2_k and k ^ 1.3 [89]. The contribution 
of the extrinsic base resistance noise becomes more important with scaling, especially for the device 
with very high transition frequency / T [90]. 



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ACTIVE DEVICES AND MODELING 



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ACTIVE DEVICES AND MODELING 



57. A. van der Ziel, "Thermal Noise in the Hot Electron Regime in FET's," IEEE Trans. Electron Devices, vol. 
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58. R. A. Pucel, D. F. Masse, and C. F. Krumm, "Noise Performance of Gallium Arsenide Field-Effect Transistors," 
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3 IMPEDANCE MATCHING 



This chapter begins by describing the main principles and impedance matching tools. Generally, an 
optimum solution depends on the circuit requirement, such as the simplicity in practical realization, 
the frequency bandwidth and minimum power ripple, design implementation and adjustability, stable 
operation conditions, and sufficient harmonic suppression. As a result, many types of the matching 
networks are available, including lumped elements and transmission lines. To simplify and visualize 
the matching design procedure, an analytical approach, which allows calculation of the parameters of 
the matching circuits using simple equations, and Smith chart traces are discussed. In addition, several 
examples of the narrowband and broadband power amplifiers using bipolar or MOSFET devices are 
given, including successive and detailed design considerations and explanations. 



3.1 MAIN PRINCIPLES 

Impedance matching is necessary to provide maximum delivery to the load of the RF power available 
from the source. This means that, when the electrical signal propagates in the circuit, a portion of this 
signal will be reflected at the interface between the sections with different impedances. Therefore, it is 
necessary to establish the conditions that allow to fully transmit the entire electrical signal without any 
reflection. To determine an optimum value of the load impedance Zl, at which the power delivered 
to the load is maximal, consider the equivalent circuit shown in Figure 3.1(a). 
The power delivered to the load can be defined as 



where Z s = R s + jX s is the source impedance, Z L = i? L +jX L is the load impedance, V s is the source 
voltage amplitude, and K in is the load voltage amplitude. Substituting the real and imaginary parts of 
the source and load impedances Zs and Zl, into Eq. (3.1) yields 



Assume the source impedance Z s is fixed and it is necessary to vary the real and imaginary parts 
of the load impedance Zl until maximum power is delivered to the load. To maximize the output 
power, the following analytical conditions in the form of derivatives with respect to the output power 
are written: 




(3.1) 



1 

P = - 



2 s (R s + R L f + (X s + X L ) 



(3.2) 



BP dP 
= 0 = 0. 



(3.3) 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



113 



114 IMPEDANCE MATCHING 





la) 



o- 

iL 



z in -z. 



(b) 

FIGURE 3.1 Equivalent circuits with (a) voltage and (b) current sources. 



Applying these conditions and taking into consideration Eq. (3.2), the following system of two 
equations can be obtained: 



1 



2R L (R L + R S ) 



(R L + R s ) 2 + (X L + X s ) 2 [ ( fl L + Rs f + (x L + X s ) 2 f 

2X L (X L + X s ) 



0 



[(H L + ^s) 2 + (X L + X s ) 2 ] 

Simplifying Eqs. (3.4) and (3.5) results in 

R 2 S - Rl + (X L + X s ) 2 = 0 
X L (X L + X S ) = 0. 

By solving Eqs. (3.6) and (3.7) simultaneously for R s and X s , one can obtain 

^s = 

Xl = — X s 



or in a common impedance case 



Z L — Zg 



(3.4) 
(3.5) 



(3.6) 
(3.7) 



(3.8) 
(3.9) 

(3.10) 



where * denotes the complex-conjugate value [1]. 

Eq. (3.10) is called an impedance conjugate matching condition, and its fulfillment results in 
maximum power delivered to the load for fixed source impedance. 



MAIN PRINCIPLES 



115 



Maximum power delivered to the load must be equal to 




(3.11) 



The admittance conjugate matching condition, applied to the equivalent circuit presented in 
Figure 3.1(b), 



can be readily obtained in the same way. Maximum power delivered to the load in this case can be 
written as 




(3.13) 



where Gs = ReFs is the source conductance and / s is the source current amplitude. 

Thus, the conjugate matching conditions in a common case can be determined through the 
immittance parameters, that is any system of impedance Z-parameters or admittance y-parameters, 
in the form of 



The matching circuit is connected between the source and the input of an active device shown 
in Figure 3.2(a) and between the output of an active device and the load shown in Figure 3.2(b). 



(3.12) 



w h = w*. 



(3.14) 





L 



0) 




t 

o 



t 

o 



-o 



FIGURE 3.2 Matching circuit arrangements. 



116 IMPEDANCE MATCHING 



For a multistage power amplifier, the load represents an input circuit of the next stage. Therefore, 
the matching circuit is connected between the output of the active device of the previous amplifier 
stage and the input of the active device of the subsequent stage of the power amplifier shown in 
Figure 3.2(c). The main objective is to properly transform the load immittance to the optimum 
device output immittance W out , the value of which is properly determined by the supply voltage, the 
output power, the saturation voltage of the active device, and the selected class of the active device 
operation to maximize the operating efficiency and output power of the power amplifier. It should be 
noted that Eq. (3 . 14) is given in a general immittance form without indication of whether it is used in a 
small-signal or large-signal application. In the latter case, this only means that the device immittance 
W-parameters are fundamentally averaged over large signal swing across the device equivalent circuit 
parameters and that the conjugate-matching principle is valid in both the small-signal application and 
the large-signal application, where the optimum equivalent device output resistance (or conductance) 
is matched to the load resistance (or conductance) and the effect of the device output reactive elements 
is eliminated by the conjugate reactance of the load network. In addition, the matching circuits should 
be designed to provide the required voltage and current waveforms at the device output and the 
stability of operation conditions, and according to the requirements for the power amplifier amplitude 
and phase characteristics. The losses in the matching circuits must be as small as possible to deliver 
the output power to the load with maximum efficiency. Finally, it is desirable that the matching circuit 
be easy to tune. 



3.2 SMITH CHART 

The hemisphere Smith chart is one of the tools most widely used to match circuit designs because it 
gives a clear and simple graphical representation of the consecutive matching design procedure [2,3]. 
However, another hemisphere chart with a standing-wave indicator to measure the impedance ratio 
of a load on line where a grid of circular lines was marked with magnitude and angle of impedance, 
corresponding to latitude and longitude on a hemisphere, was simultaneously proposed by Carter 
[4,5]. The Smith chart can be applied for matching using both lumped elements and transmission 
lines. The Smith chart is particularly useful for matching circuit designs that use the transmission 
lines because analytical calculations in this case are very complicated. Also, when using the complete 
Smith chart, the circuit parameters such as voltage standing wave ratio ( VSWR), reflection coefficient, 
return loss, or losses in the transmission line can be directly calculated. The Smith chart is a very 
important tool, being a part of modern computer-aided design software and test equipment and 
providing a useful visual way to understand the circuit behavior. 

The Smith chart represents a relationship between the load impedance Z and the reflection 
coefficient T that can be written in the normalized form of 

Z 1 + r 

% = (3 - 15) 

where the normalized impedance Z/Zo can also be written as 

Z R X 

T = T + j Y (3 ' 16) 

Z.Q ZjQ Z.Q 



and represents the reflection coefficient T through its real and imaginary parts as 



r = r r + ;Ti 



(3.17) 



SMITH CHART 117 



Then, substituting Eqs. (3.16) and (3.17) into Eq. (3.15) results in 



R X 1 + T r + /Tj 

h j — = — . (3.18) 

Z„ Z 0 1 - r r - ;Ti 



By equating the real and imaginary parts of Eq. (3.18), one can obtain 

r '-^) 2 + r -(^Tzo) 2 (3 ' 19) 
(r, - D 2 + (r, - ^ ) = (y) ■ (3 ' 20) 

As a result, in the r r -r; coordinate plane, Eq. (3.19) represents a family of circles centered at 
points r r = R/(R + Zq) and rj = 0 with radii of Z 0 /(R + Z 0 ), which are called constant-(RIZ Q ) circles. 
Eq. (3.20) represents a family of circles at points r r = 1 and Tj = ZqIX with radii of Zq/X, which 
are called constant-(X/Z 0 ) circles. These constant-(W/Z 0 ) and constant-(AVZ 0 ) circles with different 
normalized parameters are shown in Figure 3.3(a), where the points r r = — 1 and r r = 1 are also 
indicated. The plot of such circles is called the impedance Smith chart or the Z Smith chart. The curve 
from the point A to the point C represents the impedance transformation from the pure resistance of 
25 Q to the inductive impedance of (25 + j25) Q, which can be provided by using the inductance 
connected in series with the resistance. 

Eqs. (3.19) and (3.20) can be rewritten easily in the admittance form with the real part G and 
imaginary part B when a relationship between the impedance Y and the reflection coefficient T can 
be written as 

G B 1 - r r - ;Ti 

+ jV= , ,p , V (3- 21 ) 



y 0 " y 0 i + r r + jri 



r r + ~ TT ) + r ? = ( TT^V I O- 22 ) 



where Y 0 = 1/Z 0 . Then 



G + Y 0 J 1 \G + Y 0 
(r r +l) 2 +(r i+ ^) 2 =(!) 2 . (3.23) 

From Eqs. (3.22) and (3.23) it follows that the constant-(G/l'o) circles are centered at T r = 
— G/(G + Y 0 ) and T; = 0 with radii of Y 0 I(G + Y 0 ). The constant-(B/Y 0 ) circles are centered at 
points r r = — 1 and V; — —Y 0 /B with radii of Y 0 /B, which is shown in Figure 3.3(b). These circles 
are centered at antisymmetric points in contrast to the impedance Smith chart. The admittance Smith 
chart or the Y Smith chart, the admittances of which coincide with the appropriate impedances 
plotted at the Z Smith chart, is the mirror-reflected impedance Smith chart as a result of its rotation 
by 180°. The curve from the point C to the point D shows the admittance transformation from the 
inductive admittance of (20 — j2Q) mS to the pure conductance of 20 mS or resistance of 50 Q, 
which can be provided by using the capacitance connected in parallel with the initial admittance. The 
impedances and admittances can be determined by any impedance or admittance Smith chart where 
the normalized parameters are indicated. As a general case, this diagram is called the immittance 
Smith chart. However, in this case, the impedance point and its corresponding admittance value are 
located one against another at the same distance from center (1,0). 

Therefore, sometimes it is advisable to use the combined impedance-admittance Smith chart 
shown in Figure 3.3(c). This is the case when, for any point, we can read the normalized impedance 
from the Z Smith chart and normalized admittance from the Y Smith chart. This Z-Y Smith chart 



IMPEDANCE MATCHING 




SMITH CHART 119 



avoids the necessity of impedance rotating by 180° to find the corresponding admittance. A combined 
impedance-admittance Smith chart is very convenient for matching using lumped elements. For 
example, it is necessary to convert the source active impedance of 25 £2 (point A) into the load 
resistance of 50 Q (point D). First, the series inductance plotted at the Z Smith chart changes the 
source impedance by moving along constant-(X^Zo) circle from point A as far as point C. Then, it 
is converted to the Y Smith chart. As a result, the parallel capacitance starting at this point changes 
the given admittance by moving along constant-(8/Y 0 ) circle from point C as far as point D. Such 
a transformation between these two resistances for normalizing impedance Zo = 50 Q is shown in 
Figure 3.3(c). 

When designing RF and microwave power amplifiers, it is very important to determine such 
parameters as VSWR, reflection coefficient, or losses in the matching circuits based on the lumped 
elements or using the transmission lines. The linear reference scales, indicating these additional 
characteristics, are added underneath the Smith chart, as shown in Figure 3.4. Scales around its 




FIGURE 3.4 Smith chart. 



120 IMPEDANCE MATCHING 



periphery show the calibrated electrical wavelength and the angles of the reflection coefficients. 
The Smith chart can be easily implemented for the graphical design using the transmission lines. 
Expression for the input impedance of the lossless transmission line can be written in terms of the 
reflection coefficient as 

= l + rexp(-2j0) 
Z 0 1 - rexp(-2j0)' 

This equation differs from Eq. (3.15) by the added phase angle only. This means that the normalized 
input impedance, seen looking into the transmission line with electrical length 6, can be found by 
rotating this impedance point clockwise by 26 about the center of the Smith chart with the same 
radius | T | . By subtracting 26 from the phase angle of the reflection coefficient, its value decreases 
in the clockwise direction, according to the periphery scale. In this case, the half-wave transmission 
line provides a clockwise rotation of 27r or 360° about the center, returning the point to its original 
position. 

For example, a typical 50 Q transmission line provides a transformation of load impedance from 
Z L = (12 + jl0) Q to a new impedance of Z in = (100 +7 100) Q. The normalized load impedance 
is Z L /Z 0 = (0.24 + j0.2) Q, which is plotted on the Smith chart, as shown in Figure 3.4. By using 
a compass to draw the circle from this point to the intersection point with a real axis, we obtain 
|T| = 0.61 at the reflection coefficient scale, RL — 4.3 dB at the return loss scale and VSWR — 4.2 
at the standing wave ratio (SWR) scale. To determine the angle of the reflection coefficient, it needs 
to draw a radial line through the load impedance point to the intersection of the periphery circle 
(an angle of 83° can be read). If a radial line is drawn through the point of the input impedance 
at the outer wavelength scale, the difference between points of 0.209A and 0.033X gives the length 
of the transmission line as 0.176A. However, in the case of the transmission line with losses of 
2 dB, the point obtained should be moved to the point of Z m — (90 + _/40) £2, according to the 
attenuation scale. 



3.3 MATCHING WITH LUMPED ELEMENTS 
3.3.1 Analytic Design Technique 

Generally, there is a variety of configurations for matching networks that can be used to connect a 
generating system efficiently to its useful load, but in order to obtain high transmission efficiency, 
any of these networks should be properly designed. The lumped matching circuits in the form of (a) 
L-transformer, (b) it -transformer, or (c) L-transformer shown in Figure 3.5 have proved for a long 
time to be effective for power amplifier design [6]. The simplest and most popular matching circuit 
is the matching circuit in the form of the L-transformer. The transforming properties of this matching 
circuit can be analyzed by using the equivalent transformation of a parallel into a series representation 
of RX circuit [7]. 

Consider the parallel RX circuit shown in Figure 3.6(a), where R { is the real (resistive) and Xj 
is the imaginary (reactive) parts of the circuit impedance. Zj = jXiRi/(Ri + jX\) and the series 
RX circuit shown in Figure 3.6(b), where R2 is the resistive and X 2 is the reactive parts of the circuit 
impedance Z 2 — R2 + jX 2 . These two circuits, series and parallel, can be considered equivalent at 
some frequency if Z\ — Z 2 resulting in 



1 



(a) 



MATCHING WITH LUMPED ELEMENTS 



111 



J I 

(b) 



T o 



(c) 

FIGURE 3.5 Matching circuits in the form of (a) L-, (b) it- and (c) ^-transformers. 



ft 




ft. 



(a) (*) 

FIGURE 3.6 Impedance (a) parallel and (b) series equivalent circuits. 



Eq. (3.25) can be rearranged by two separate equations for real and imaginary parts in the form of 



Ri =Ri(l + Q 2 ) 
X, = X 2 (1 + Q- 2 ) 



(3.26) 
(3.27) 



where Q — Ri/\X[ \ — \X 2 \/R2 is the circuit quality factor, which is equal for both the series and parallel 
RX circuits. 

Consequently, if the reactive impedance Xi = — X2 (l + Q~ 2 ) is connected in parallel to the series 
circuit R2X2, it allows the reactive impedance (or reactance) of the series circuit to be compensated. 
In this case, the input impedance of such a two-port network shown in Figure 3.7 will be resistive 



Z„, - R 



1 — 0- 



X, 



FIGURE 3.7 Input impedance of two-port network. 



122 IMPEDANCE MATCHING 




co L, = Q R 2 J \ tf 2 ' od C, =1/(QR,) 

FIGURE 3.8 L-type matching circuits and relevant equations. 



only and equal to . Consequently, to transform the resistance R[ into the other resistance R 2 at the 
given frequency, it is sufficient to connect between them a two-port L-transformer with the opposite 
signs of the reactances X { and X 2 , the parameters of which can be easily calculated from the following 
simple equations: 

\X 2 \ = R 2 Q (3.29) 



where 




(3.30) 



is the circuit quality factor expressed through the resistances to be matched. Thus, to design a 
matching circuit with fixed resistances to be matched, first we need to calculate the circuit quality 
factor Q according to Eq. (3.30) and then to define the reactive elements, according to Eqs. (3.28) 
and (3.29). 

Due to the opposite signs of the reactances X[ and X 2 , the two possible circuit configurations (one 
in the form of a low-pass filter section and another in the form of a high-pass filter section) with 
the same transforming properties can be realized, as shown in Figure 3.8 together with the design 
equations. 

The lumped matching circuits in the form of the L-transformer loaded on the resistance R 2 can also 
be considered as the parallel resonant circuit shown in Figure 3.9. In this case, the series inductance L 2 
and series resistance R 2 transform to the parallel inductance L 2 and parallel resistance R' 2 , respectively, 
which become the frequency-dependent functions: 



R { = R' 2 = R 2 (1 + Q 2 ) 
L' 2 =L 2 {1+ Q- 2 ) 



(3.31) 
(3.32) 



MATCHING WITH LUMPED ELEMENTS 



123 



Li 




FIGURE 3.9 Parallel resonant circuit equivalent of loaded L-transformer. 



where Q — (dL%IR%. The resonant frequency /o of such an equivalent parallel resonant circuit is 
determined from 



m 0 = 2jr/ 0 = 




(3.33) 



If this matching circuit has small values of Q, wider frequency bandwidth but poor out-of-band 
suppression can be achieved. However, with large values of Q, the frequency bandwidth is substantially 
reduced. For the case of R1/R2 > 10, which corresponds to the condition of Q > 3, the frequency 
bandwidth A/ and out-of-band suppression factor F B of such an L-transformer can be approximately 
evaluated by the same formulas as for a parallel resonant circuit: 

A/ = ^ (3.34) 

F a = Q (n 2 - 1) (3.35) 

where /o is the operating frequency and n is the harmonic number [8]. The out-of-band suppression 
factor F n of the matching circuit represents the ratio of the selected harmonic component in the input 
(collector) current to the same harmonic component in the output (load) current. Figure 3.10 shows 
the frequency behavior of the input impedance magnitude IZ in l of the parallel resonant circuit. 



UJ i 



0.707/i, 




1 1 +■ 

0 1.0 jy n 

FIGURE 3.10 Frequency plot of input impedance for parallel resonant circuit. 



124 IMPEDANCE MATCHING 



The transformer efficiency r/j is determined by the ratio of P^/Pia, where P m is the power at the 
input of the transformer and P L is the load transformer power. The efficiency for the L-transformer 
with negligible losses in the capacitor can be calculated from 



where Q inA is the inductor quality factor. From Eq. (3.36) it follows that, with the increase of Q, 
the efficiency of the impedance transformer decreases. This means that, for the same R\ and series 
parasitic resistance of the circuit inductance, the lower resistance R 2 provides the higher current 
flowing through this inductance, which leads to an additional power dissipation. An analysis of 
Eqs. (3.28) and (3.29) shows that, for the given resistances Ri and R 2 , each element of the L- 
transformer can have only one value for a fixed frequency. Consequently, it is difficult to satisfy 
simultaneously such contradictory requirements as efficiency, frequency bandwidth and out-of-band 
suppression. 

To avoid the parasitic low-frequency oscillations and to increase the level of the harmonic sup- 
pression, it may be necessary to connect an additional L 0 Co series circuit, which resonant frequency 
is equal to the operating frequency of the power amplifier, as shown in Figure 3.1 1. In this case, the 
out-of-band suppression factor F n defined by Eq. (3.35) is written as 



where Qs — Q + Qo and <2o = cl>Lq/R 2 . Better harmonic suppression is achieved at the expense of 
the frequency bandwidth narrowing. 

In practice, it makes sense to use the single two-port L-transformers in power amplifiers as the 
interstage matching circuits, where the requirements for out-of-band suppression and efficiency are 
not as high as for the output matching circuits. In this case, the main advantage of such an L- 
transformer is in its simplicity when the only two reactive elements with fast tuning are needed. For 
larger values of Q > 10, it is possible to use a cascade connection of L-transformers, which allows 
wider frequency bandwidth and transformer efficiency to be realized. 

The matching circuits in the form of (a) it -transformer and (b) L-transformer can be realized by 
appropriate connection of two L-transformers, as shown in Figure 3.12. For each L-transformer, the 
resistances R t and the resistance R 2 are transformed to some intermediate resistance R 0 with the value 
of Ro < (Ri, R2) for a it -transformer and the value of Rq > (Mi, Ri) for a L-transformer. The value 
of R 0 is not fixed and can be chosen arbitrary depending on the frequency bandwidth. This means 
that, compared to the simple L-transformer with fixed parameters for the same ratio of R2/R1 , the 
parameters of the it -transformer or L-transformer can be different. However, they provide narrower 
frequency bandwidths due to higher quality factors because the intermediate resistance Rq is either 
greater or smaller than each of the resistances R t and R 2 . By taking into account the two possible 




(3.36) 



L„ = 2e (n 2 




(3.37) 




C"'„ 



a 



■Zin - R) 



O i C 

FIGURE 3.11 L-transformer with additional LC-filter. 



MATCHING WITH LUMPED ELEMENTS 



125 



-V, 



L-L 



n 



(a) 




FIGURE 3.12 Matching circuits developed by connecting two L-transformers. 



circuit configurations of the L-transformer shown in Figure 3.8, it is possible to develop the different 
circuit configurations of such two-port transformers shown in Figure 3.12(a), where X$ = X' 3 + X' 3 \ 
and in Figure 3.12(6), where X 3 = X' 3 X'{/ (X' 3 + X%). 

Several of the most widely used two-port it -transformers, together with the design formulas, 
are shown in Figure 3.13 [7,9]. The it -transformers are usually used as output matching circuits of 
high-power amplifiers in class B operation when it is necessary to achieve a sinusoidal drain (or 
collector) voltage waveform by appropriate harmonic suppression. In addition, it is convenient to use 
some of them as interstage matching circuits in low-power and medium-power amplifiers when it is 
necessary to provide sinusoidal voltage waveforms both at the drain (or collector) of the previous 
transistor and at the gate (or base) of the subsequent transistor. In this case, for the it -transformer 
with shunt capacitors, the input and output capacitances of these transistors can be easily included 
into the matching circuit elements Ci and C 2 , respectively. Finally, a it -transformer can be directly 
used as the load network for a high-efficiency class E mode with proper calculation of its design 
parameters. 

Figure 3.14 shows the it -transformer with shunt capacitor and additional L 0 Co series circuit in 
which elements are defined from 



coLn = 



1 

wC 0 



gigo 

i + Qi 



(3.38) 



where Q 0 is the arbitrary value depending on the specification requirements to the harmonic suppres- 
sion. The other elements are defined by the design equations given in Figure 3.13(a). 

The it -transformer with the two shunt capacitors, shown in Figure 3.13(a), represents a face-to- 
face connection of two simple low-pass L-transformers. As a result, there is no special requirement 
for the resistances Ri and R 2 in which ratio R l IR 2 can be greater or smaller than unity. As an example, 
the design equations correspond to the case of R1IR2 > 1. However, as it will be further derived, 
the ?r -transformer with a series capacitor shown in Figure 3.13(b) can only be used for impedance 
matching when RJR 2 > 1 . Such a ^-transformer represents a face-to-face connection of the high-pass 
and low-pass L-transformers, as shown in Figure 3.15(a). 



IMPEDANCE MATCHING 



0) C, = (), /R, CO C", = (), l&~ £0 £, = /{,((), I ft)/(l I 

ft = ;' ffj (i + o, J )- 1 ft > R ' - l 

(a) 
('."■> 



C 2 d= #2 



0) A, = R, /(?, « ft = ft i ff, co ft = (l + ftX^fo 



a, 



<o = /?, /ft o c; = ft / to = (ft - o,)/(i + 



a -^5 (i -of)-' 

(e) 

FIGURE 3.13 jt -transformers and relevant equations. 



Lt 



O * *— 

FIGURE 3.14 7r -transformer with additional LC-filter. 



MATCHING WITH LUMPED ELEMENTS 



127 



C3 U 




(b) 



FIGURE 3.15 jt -transformer with series capacitor. 



The design equations for a high-pass section are written using Eqs. (3.28) to (3.30) as 



a>U = -r- (3.39) 
Hi 

a>C 3 = (3.40) 
QiRo 

Q\ = ~ - 1 (3.41) 



where <2i is the circuit quality factor and R 0 is the intermediate resistance. 
Similarly, for a low-pass section, 



Qz 



(oL' 3 — Q 2 Ro 



(3.42) 
(3.43) 
(3.44) 



Since it is assumed that Ri > R 2 > Ro, from Eq. (3.41) it follows that the quality factor Q\ of a 
high-pass L-transformer can be chosen from the condition of 



n R\ 



(3.45) 



128 IMPEDANCE MATCHING 



Substituting Eq. (3.41) into Eq. (3.44) results in 



+ (3.46) 

Combining the reactances of two series elements (capacitor C' 3 and inductor L' 3 ) given by Eqs. 
(3.40) and (3.43) yields 

^-—; = R 0 (Q 2 - Gi) = 7. ' 2 f ■ (3 ' 47) 
coC 3 (1 + Ql) 

As a result, since Q\ > Q 2 , the total series reactance is negative, which can be provided by a series 
capacitance C3, as shown in Eq. 3.15(b), with susceptance 

1 + Ql 

a>C 3 = — . (3.48) 

Ri(Qi-Qi) 

On the other hand, if Q 2 > Q\ when R 2 > R\> Ro, then the total series reactance is positive, which 
can be provided by a series inductance L 3 , and all matching circuit parameters can be calculated 
according to the design equations given in Figure 3.13(c). In this case, it needs first to choose the 
value of Q 2 for fixed resistances R 1 and R 2 to be matched, then to calculate the value of gi, and finally 
the values of the shunt inductance L\, shunt capacitance C 2 , and series inductance L3. 

Some of the matching circuit configurations of two-port T-transformers, together with the design 
formulas, are shown in Figure 3.16 [7,9]. The T-transformers are usually used in the high-power 
amplifiers as input, interstage, and output matching circuits, especially the matching circuit with 
shunt and series capacitors shown in Figure 3.16(b). In this case, if a high value of the inductance 
Li is chosen, then the current waveform at the input of the transistor with a small input resistive 
part will be close to sinusoidal. By using such a r-transformer for the output matching of a power 
amplifier, it is easy to realize a high-efficiency class F operation mode, because the series inductance 
connected to the drain (or collector) of the active device creates an open-circuit harmonic impedance 
conditions. 

The r-transformer with the two-series inductors, shown in Figure 3.16(a), represents a back-to- 
back connection of two simple low-pass /.-transformers. In this case, the resistance ratio ^1/^2 can 
be greater or smaller than unity, similar to a it -transformer with the two shunt capacitors shown in 
Figure 3.13(a). As an example, the design equations correspond to the case of R[IR 2 > 1. However, 
the T-transformer with series and shunt capacitors shown in Figure 3.16(b) can only be used for 
impedance matching when RJR 2 > 1. Such a r-transformer represents a back-to-back connection of 
the high-pass and low-pass L-transformers, as shown in Figure 3.17(a). 

The design equations for a high-pass section of such a r-transformer are written using Eqs. (3.28) 
to (3.30) as 

a> Cl = ^ (3.49) 

coL' 3 = ^ (3.50) 

Q\ = J - 1 (3.51) 
R\ 



where Q\ is the circuit quality factor and Rq is the intermediate resistance. 



MATCHING WITH LUMPED ELEMENTS 129 



Li 



1 



0 + a)- 1 y;> J - 1 

(a) 



f: 3 



^3 



, c, = i /(», a ) 0) r.j = £>, fl j <» c 3 = . - Q, y\jt,. (i + qM 



(J, 



(ft) 



«1 



o- 



, c, = 1 /(*,e,) ft) r.j = » ^ = + 



FIGURE 3.16 r-transformers and relevant equations. 



Similarly, for a low-pass section, 



a)L 2 = Q 2 R 2 
uC 3 = Q2/R0 



Q 



2 ^0 



Rn 



L. 



(3.52) 
(3.53) 

(3.54) 



Since it is assumed that R 0 > i?i > R 2 , from Eq. (3.54) it follows that the quality factor Q 2 of a 
low-pass L-transformer can be chosen from the condition of 



o R\ 



(3.55) 



130 IMPEDANCE MATCHING 




00 

u 



FIGURE 3.17 //-transformer with series and shunt capacitors. 



Substituting Eq. (3.54) into Eq. (3.51) results in 



| (i + Ql) 



(3.56) 



Combining the susceptances of two shunt elements (inductor L' 3 and capacitor C' 3 ) given by Eqs. 
(3.50) and (3.53) yields 



62-81 
R 0 



62-61 

Ri (1 + el) ' 



(3.57) 



As a result, since Qi> Q\, the total shunt susceptance is positive, which can be provided by a 
shunt capacitance C3, as shown in Figure 3.17(b), with susceptance 



C0C3 



62-61 

fl 2 (1 + el) ' 



(3.58) 



On the other hand, if gi > 62 when R 0 > R 2 > Ri, then the total shunt susceptance is negative 
that can be provided by a shunt inductance L3, and all matching circuit parameters can be calculated 
according to the design equations given in Figure 3.16(c). In this case, it needs first to choose the 
value of Qi for fixed resistances Ri and Ri to be matched, then to calculate the value of Q2, and finally 
the values of the series capacitance C\, series inductance Li, and shunt inductance L3. 

If the elements of it- and T-transformers are chosen according to the condition X\ — X 2 — —X 3 , 
then the input impedance Z- m of the transformer loaded by the resistance R^ (from any side) is equal 
to 



X 2 
7^ 



(3.59) 



MATCHING WITH LUMPED ELEMENTS 



131 



where X — i — 1, 2, 3. As a result, the input impedance Z in will be resistive, regardless of the value 
of the load resistance R L . For example, setting R L — R 2 for the transformers shown in Figure 3.13(a) 
and Figure 3.16(a) yields 



When Xi ^ X 2 ^ — X-j, the input impedances of the it- and T-transformers will be resistive for 
only one particular value of R^. 

3.3.2 Bipolar UHF Power Amplifier 

Consider a design example of a 10 W 300 MHz bipolar power amplifier with supply voltage 12.5 V 
providing a power gain of at least 10 dB. The first design step is to select an appropriate active device 
that allows both simplifying the circuit design procedure and satisfying the specified requirements. 
Usually, the manufacturer states the values of the input and output impedances or admittances at the 
nominal operation point on the data sheet for the device. For example, the above requirements can 
be realized by an n-p-n silicon bipolar transistor operating at 300 MHz with the input impedance 
Zi„ = (1.3 + j0.9) £2 and output admittance Y out — (150 — /70) mS, which is intended for power 
amplification in class AB with nominal supply voltages up to 13.5 V. 

In this case, Z in is expressed as a series combination of the transistor input resistance and inductive 
reactance, whereas K out is represented by a parallel combination of the transistor output resistance and 
inductive reactance. This means that, at required operating frequency, effect of the series parasitic 
collector lead inductance exceeds the effect of the shunt collector capacitance, which gives a net 
inductive reactance to the equivalent output circuit of an active device. To match the series input 
inductive impedance to the standard 50 Q input source impedance, it is advisable to use a matching 
circuit in the form of a T-transformer shown in Figure 3.16(fc), where the series capacitance C\ can 
serve also as a blocking capacitor. Figure 3.18 shows the complete input network including input 
device elements and matching circuit. From the reactive part of the input impedance Z in it follows 
that, at the operating frequency of 300 MHz, the input inductance will be equal approximately to 
0.5 nH. 

It is necessary first to calculate the quality factor Q 2 , which is needed to determine the parameters 
of the matching circuit: 



Consequently, the value of Q 2 must be larger than 6.1. For example, Q 2 — 6.5 provides a 3-dB 
bandwidth of 300 MHz/6.5 = 46 MHz. In this case, a value of gi will be equal to 0.35. As a result, 



*i = 



X 2 



(3.60) 





a 



FIGURE 3.18 Complete input network circuit. 



132 IMPEDANCE MATCHING 



the values of the input matching circuit elements are as follows: 

d = = 30pF 

coQiRi 

L] + L m = ¥L-!± = 4.5nH=> L x = 4.0nH 

(O 

C 2 = , ^ = 59pF. 

««to (i + el) 

This type of a T-transformer is widely used in practical matching circuit design because of its 
simplicity and convenience. In addition, the sufficiently small value of a series capacitance Ci can 
contribute to the elimination of the low-frequency parasitic oscillations in the case of a multistage 
power amplifier. The function of each element can be graphically traced on the Smith chart as shown 
in Figure 3.19. 

The easiest and most convenient way to plot the traces of the matching circuit elements is by 
plotting initially the traces of Q 2 — 6.5 and <2i = 0.35. The circle of equal Q is plotted, taking into 
account that, for each point located at this circle, a ratio of XIR or BIG must be the same. The trace 
of the series inductance Li must be plotted as far as the intersection point with gi-circle. This means 




"' A ' " ' U ~ 

FIGURE 3.19 Smith chart with elements from Figure 3.18. 



MATCHING WITH LUMPED ELEMENTS 



133 



that, beginning at Z in , a curve of increasing inductive reactance must be plotted up to <22-circle. The 
value of L\ is determined from the normalized inductive impedance at this intersection point. Then, 
due to a normalization to 50 Q, the chart value must be multiplied by factor 50. 

The trace of the parallel capacitance Ci must be plotted using admittance circles. The previous 
impedance point located at (22-circle is converted to its appropriate admittance one. This point is 
symmetrical to the impedance point regarding the center point and is located on a straight line from 
the intersection point drawn through the center of the Smith chart into its lower half at the same 
distance from the center point. A curve from this point with constant conductance and increasing 
capacitive susceptance is plotted. These points are transformed to appropriate impedances using a line 
through the center point extended at equal distance on the other side and stop when the transformed 
curve reaches the <2i = 0.35 circle. In other words, it is necessary to transform mentally or to use 
a transparent admittance Smith chart (impedance Smith chart rotated on 180°) to plot a curve for 
C2 on the upper half of the impedance Smith chart. The difference between the susceptances at the 
beginning and the end of this curve determines the value of Cj- Then, a curve of reducing inductive 
reactance is plotted down to the center point to determine a value of the series capacitance Ci. 

A similar design philosophy can be applied to the design of the output matching circuit shown in 
Figure 3.20. However, taking into account the presence of the output shunt inductance, it makes sense 
to use a matching circuit in the form of a it -transformer shown in Figure 3.13(c). The equivalent 
output resistance can be analytically evaluated by 



[V cc - V cc(sat) ] 2 _ (0.9 Fee) 2 



2P„, 



2i> 



= 6.3 Q, 



where V cc is the supply voltage, V CC ( S at) is the saturation voltage, and P 0M is the output power. Its 
value is very close to the measurement value given by the real part of the device output admittance 
calculated as R out — 1/0.15 = 6.7 f2. A value of L out is approximately equal to 7.6 nH. 
The quality factor Q2 can be chosen as 



g 2 >j-jr- 1 = 2.5. 



The calculated quality factor £>i of the device output circuit is 



Qi = 



a>L„ 



= 0.47. 



This value of L out allows the output device admittance to be matched to 50 Q load using such a 
ji -transformer because 



£2 = J-^(l + e 2 )-l = 2.8>2.5. 




FIGURE 3.20 Complete output network circuit. 



134 IMPEDANCE MATCHING 



As a result, the values of the other two elements of the output matching circuit are 



31 pF 



u = 



62 

C0R2 

RiiQi-Qi) 
<b(1+01) 



= 6.8 nH. 



A blocking capacitor that performs dc supply decoupling can be connected after the it -transformer 
with sufficiently high value of its capacitance to avoid any negative influence on the matching circuit. 
Alternatively, it can be used in series with the inductor L 2 , in order to form a series resonant circuit, as 
shown in Figure 3.11. The design of the output circuit using the Smith chart is given in Figure 3.21. 
Initially, it is necessary to transform the output admittance y out to the output impedance Z out using the 
straight line of the Smith chart, putting the Z ou( point at the same distance from the center point as for 
the y out point. Then, the effect of increasing series inductance L 2 , by moving from the Z out point along 
the curve of the constant R and increasing X until intersection with the Q2 — 2.8 circle, is plotted. 
To determine the parallel capacitance C3, it is necessary to transform this point to the corresponding 




FIGURE 3.21 Smith chart with elements from Figure 3.20. 



MATCHING WITH LUMPED ELEMENTS 



135 



admittance one and plot the curve of the constant G and increasing B, which must intersect the center 
point of the Smith chart. 



3.3.3 MOSFET VHF High-Power Amplifier 

Now let us demonstrate lumped matching circuit technique to design a 150 W MOSFET power 
amplifier with the supply voltage 50 V operating in a frequency bandwidth of 132 to 174 MHz 
and providing a power gain greater than 10 dB. These requirements can be satisfied using a silicon 
rc-channel enhancement mode VDMOS transistor designed for power amplification in the VHF 
frequency range. In this case, the center bandwidth frequency is equal to f c — a/132 • 174= 152MHz. 
For this frequency, the manufacturer states the following values of the input and output impedances: 
Z in = (0.9 — jl.2) Q and Z out = (1.8 + y'2.1) Q. Both Z in and Z oul represent the series combination 
of an input or output resistance and a capacitive or inductive reactance, respectively. To cover the 
required frequency bandwidth, the low-g matching circuits should be used that allows reduction of 
the in-band amplitude ripple and improvement of the input VSWR. The value of a quality factor for 
3-dB bandwidth level must be less than Q — 152/(174- 132) = 3.6. As a result, it is very convenient to 
design the input and output matching circuits using the simple L-transformers in the form of low-pass 
and high-pass filter sections with a constant value of Q [10]. 

To match input series capacitive impedance to the standard 50 Q source impedance in a sufficiently 
wide frequency bandwidth, it is preferable to use three filter sections shown in Figure 3.22. From the 
negative reactive part of the input impedance Z m it follows that the input capacitance at the operating 
frequency of 152 MHz is equal to approximately 873 pF. To compensate at the center bandwidth 
frequency for this capacitive reactance, it is sufficient to connect in series to it an inductance of 1 .3 nH. 
Now, when the device input capacitive reactance is compensated, the design of the input matching 
circuit can proceed. To simplify the matching design procedure, it is best to cascade L-transformers 
with equal value of Q. Although equal Q values are not absolutely necessary, this provides a convenient 
guide for both analytical calculation of the matching circuit parameters and the Smith chart graphical 
design. 

In this case, the following ratio can be written for the input matching circuit: 

R\ Ro R*>. 

-B~ = — = — (3- 61 ) 

«2 ^3 ^in 

resulting in R 2 — 13 Q and i? 3 = 3.5 f2 for R SO urce — Ri — 50 Q and R ir , — 0.9 Q. Consequently, a 
quality factor of each L-transformer is equal to Q — 1.7. The elements of the input matching circuit 
using the formulas given in Figure 3.8 can be calculated as L\ = 31 nH, C\ — 47 pF, L 2 = 6.2 nH, 
C 2 = 137 pF, 1^ = 1.6 nH, C 3 = 509 pF. 

This equal-g approach significantly simplifies the matching circuit design using the Smith chart. 
When calculating a value of Q, it is necessary to plot a circle of equal Q values on the Smith 
chart. Then, each element of the input matching circuit can be readily determined, as it is shown 
in Figure 3.23. Each trace for the series inductance must be plotted until the intersection point with 



C, L 2 /, 3 + /. in f.'j„ 




FIGURE 3.22 Complete broadband input network circuit. 



136 IMPEDANCE MATCHING 




FIGURE 3.23 Smith chart with elements from Figure 3.22. 



g-circle, whereas each trace for the parallel capacitance should be plotted until intersection with 
horizontal real axis. 

To match output series inductive impedance to the standard 50 Q load impedance, it is sufficient 
to use only two filter sections, as shown in Figure 3.24. At the operating frequency of 152 MHz, the 
transistor series output inductance is equal to approximately 2.2 nH. This inductance can be used as 
a part of L-transformer in the form of a low-pass filter section. For an output matching circuit, the 
condition of equal-<2 values gives the following ratio: 

R? i?i 

t = it (3 - 62) 

"1 "out 

with the value of Ri — 9.5 £1 for ^i oa d = R2 — 50 £2 and R out — 1.8 Q. Consequently, a quality factor 
of each L-transformer is equal to Q — 2.1, which is substantially smaller than a value of Q for 3-dB 
bandwidth level. Now it is necessary to check a value of a series inductance of the low-pass section, 
which must exceed the value of 2.2 nH for correct matching procedure. The appropriate calculation 



MATCHING WITH LUMPED ELEMENTS 



137 




FIGURE 3.24 Complete broadband output network circuit. 



gives a value of total series inductance L4 + L out of approximately 4 nH. As a result, the values of the 
output matching circuit elements are L 4 = 1.8 nH, C4 = 231 pF, C5 = 52 pF, L 5 = 25 nH. 

The output matching circuit design using the Smith chart with constant g-circle is shown in 
Figure 3.25. For the final high-pass section, a trace for the series capacitance C5 must be plotted until 
the intersection with Q — 2. 1 circle, whereas a trace for the shunt inductance L s should be plotted 
until the intersection with the center point of the Smith chart. 




FIGURE 3.25 Smith chart with elements from Figure 3.24. 



138 IMPEDANCE MATCHING 



3.4 MATCHING WITH TRANSMISSION LINES 
3.4.1 Analytic Design Technique 

At very high frequencies, it is very difficult to implement lumped elements with predefined accuracy 
in view of a significant effect of their parasitic parameters, for example, the parasitic interturn and 
direct-to-ground capacitances for lumped inductors and the stray inductance for lumped capacitors. 
However, these parasitic parameters become a part of a distributed LC structure such as a transmission 
line. In this case, for a microstrip line, the series inductance is associated with the flow of current 
in the conductor and the shunt capacitance is associated with the strip separated from the ground 
by the dielectric substrate. If the line is wide, the inductance is reduced but the capacitance is large. 
However, for a narrow line, the inductance is increased but the capacitance is small. 

Figure 3.26 shows an impedance matching circuit in the form of a transmission-line transformer 
connected between the source impedance Z s and load impedance Z L . The input impedance as a 
function of the length of the transmission line with arbitrary load impedance is 



Z L + j Z 0 tan t 
1 Z 0 + j Z L tan I 



(3.63) 



where Zo is the characteristic impedance, 6 — f}I is the electrical length of the transmission line, 
P = - ^/MrSr is the phase constant, c is the speed of light in free-space, fi T is the substrate permeability, 
e r is the substrate permittivity, co is the radial frequency, / is the geometrical length of the transmission 
line [3,11]. 

For a quarter-wavelength transmission line when 6 — tt/2, the expression for Z in simplified to 



Zin — Zq/Zl 



(3.64) 



from which it follows that, for example, a 50 f2 load is matched to a 12.5 Q source with characteristic 
impedance of 25 f2. 

Usually, such a quarter-wavelength impedance transformer is used for impedance matching in a 
narrow-frequency bandwidth of 10 to 20%, and its length is chosen at the bandwidth center frequency. 
However, using a multisection quarterwave transformer widens the bandwidth and expands the choice 
of the substrate to include materials with high dielectric permittivity, which reduces the transformer's 
size. For example, by using a transformer composed of seven quarter-wavelength transmission lines 
of different characteristic impedances, whose lengths are selected at the highest bandwidth frequency, 
the power gain flatness of ± 1 dB was achieved over frequency range of 5 to 10 GHz for a 15 W GaAs 
MESFET power amplifier [12]. 

To provide a complex-conjugate matching of the input transmission line impedance Z in with the 
source impedance Zs = Rs + jX$ when R$ — ReZ;„ and.Xs = — ImZj n , Eq. (3.63) can be rewritten as 



RL + j(X L + Zptang) 
'Z 0 - X L tan# + jR L tan 6 ' 



(3.65) 



r 



Z,„ 

-o- 



z„.e 



-o— r 



FIGURE 3.26 Transmission-line impedance transformer. 



MATCHING WITH TRANSMISSION LINES 



139 



For a quarter- wavelength transformer, Eq. (3.65) can be divided into two separate equations 
representing the real and imaginary parts of the source impedance Z s as 

«L + X L 

X s = -Zl 7 Xl . . (3.67) 
°Rl + Xl 

For a pure real load impedance with X L — 0, a quarter-wavelength transmission line with the 
characteristic impedance Zo can provide impedance matching for a purely active source and load 
only when 

Z 0 = y/R s R L . (3.68) 

Generally, Eq. (3.65) can be divided into two equations representing the real and imaginary parts, 

R s (Z 0 - X L tmO) - R L (Z 0 - X s tan 9) = 0 (3.69) 
X s (X L tan0 - Z Q )- Z 0 (X l + Z 0 tan 9) + R S R L tan 9 = 0. (3.70) 

Solving Eqs. (3.69) and (3.70) for the two independent variables Zq and 9 yields 

z 0 = jMJg + ^-Mag+ja (3 . 71) 

. / Rs- Rl \ 
0 = tan- 1 (Z o — . (3.72) 

\ ^S^L — XsRlJ 

As a result, the transmission line with the characteristic impedance Zo and electrical length 9, 
determined by Eqs. (3 .7 1 ) and (3 .72), can match any source and load impedances when the impedance 
ratio gives a positive value inside the square root in Eq. (3.71). 

For a particular case of a purely active source when Z s — Rs, the ratio between the load and 
transmission-line parameters can be expressed by 

X L Z Q (1 - tan 2 9) + (Z 2 - X\ - R[) tan9 = 0. (3.73) 

Then, for the electrical length of the transmission line having 9 — jt/4, the expression for the 
characteristic impedance Zq given by Eq. (3.71) can be simplified to 



Zo = \Z L \ = y/Rl + Xl (3.74) 
whereas the required real source impedance Rs should be equal to 

Zo 

Rs = tf L — ■ (3.75) 

Zo-X L 

Consequently, any load impedance can be transformed to a real source impedance defined by 
Eq. (3.75) using a A/8 transformer, the characteristic impedance of which is equal to the magnitude 
of the load impedance [13]. 



140 IMPEDANCE MATCHING 



Zoi. A/8 



I 



Zo 2 , A/4 Zoj. A/8 

I -O H 



•"I 



FIGURE 3.27 Transmission-line transformer for any source and load impedances. 



Applying the same approach to match purely resistive load with the source impedance, the total 
matching circuit that includes two A/8 transformers and a X/4 transformer can provide impedance 
matching between any complex source impedance Zs and load impedance Zl, as shown in Figure 3.27. 

In practice, to simplify the power amplifier design at microwaves, the simple matching circuits 
are very often used, including an L-transformer with a series transmission line as the basic matching 
section. It is convenient to analyze the transforming properties of this matching circuit by substituting 
the equivalent transformation of the parallel RX circuit to the series one. For example, Ri is the 
resistance and Xi — — ilaiC is the reactance of the impedance Z\ — jR^Xi/(Ri + jX\) for a parallel 
/?C-circuit, and R- m = ReZ in is the resistance and X- m — ImZ in is the reactance of the impedance 
Z in = R m + jX in for the series transmission-line circuit shown in Figure 3.28. For a conjugate 
matching when Zi = we obtain 



RiXi 



R 2 l X l 



R 2 + X 2 + 1 R\ + X\ 



= R„ 



(3.76) 



The solution of Eq. (3.76) can be written in the form of two expressions for real and imaginary 
impedance parts by 



Ri = Rin (1 + Q 2 ) 
Xi = -X in (l + Q- 2 ) 



(3.77) 
(3.78) 



where Q = Rx/\X\\ = X m / R ia is a quality factor equal for both parallel capacitive and series 
transmission-line circuits. Using Eq. (3.63), the real and imaginary parts of the input impedance 
Z; n can be written as 



Z 0 R 2 



1 + tan 2 



X, n = Z n tan 



Z 2 + (R 2 tan6»r 
R 2 



7 2 



Z 2 + (R 2 tan6>) 2 



(3.79) 
(3.80) 



-O- 



J d i r 



Z\ z,„ 
— o — 



z„,e 



FIGURE 3.28 L-transformer with series transmission line. 



MATCHING WITH TRANSMISSION LINES 



141 



From Eq. (3.80) it follows that an inductive input impedance (necessary to compensate for the 
capacitive parallel component) is provided when Z 0 > R 2 f° r 6 < it/2 and Z 0 < ^2 f° r t/2 < 
9 < ?r. As a result, to transform the resistance Ri into the other resistance R 2 at the given 
frequency, it is necessary to connect a two-port L-transformer (including a parallel capacitance 
and a series transmission line) between them. When one parameter (usually the characteristic 
impedance Zo) is known, the matching circuit parameters can be calculated from the following two 
equations: 



C = 



sin 29 



(0R1 

2Q 
Zo R2 
R 2 Z 0 



(3.81) 
(3.82) 



where 



<2 = 



Ri 
R2 



cos 2 e + 



(3.83) 



is the circuit quality factor defined as a function of the resistances R t and R 2 and the parameters of 
the transmission line, the characteristic impedance Zo and electrical length 9. 

It follows from Eqs. (3.82) and (3.83) that the electrical length 9 can be calculated as a result of 
the numerical solution of a transcendental equation with one unknown parameter. However, it is more 
convenient to combine these two equations and to rewrite them in the implicit form of 



#1 

#2 



1 + 



Zo 
Ri 



R2 
Z 0 



(3.84) 



Figure 3.29 shows the resistance ratio of Ri/R 2 as a function of the normalized parameter Z 0 /R 2 and 
electrical length 9 in the form of two nomographs: for the case of Z 0 /R 2 >0 shown in Figure 3.29(a) and 
for the case of Zq/R 2 < 0 shown in Figure 3.29(£>). When the input resistance Ri and output resistance 
R 2 are known in advance, it is easy to evaluate the required value of 9 for a fixed transmission-line 
characteristic impedance Z 0 using these nomographs. The graphical results show that, in contrast to a 
lumped L-transformer, a transmission-line L-transformer can match purely resistive source and load 
impedances with any ratio of Ri/R 2 . 

A n -transformer can be realized by back-to-back connection of the two L-transformers, as shown 
in Figure 3.30(a), where resistances R\ and R 2 are transformed to some intermediate resistance ./Jo- 
in this case, to minimize the length of a transmission line, the value of R 0 should be smaller than 
that of both Ri and R 2 , that is, ^0 < (Ri, ^2). The same procedure for a L-transformer shown in 
Figure 3.30(£>) gives a value of Rq that is larger than that of both Ri and R 2 , that is Rq > (Ri, R 2 ). 
Then, for a L-transformer, two shunt adjacent capacitances are combined. For a it -transformer, two 
adjacent series transmission lines are combined into a single transmission line with total electrical 
length. 

For a it -transformer, the lengths of each part of the combined transmission line can be calculated 
by equating the imaginary parts of the impedances from both sides at the reference plane A— A' to 
zero, which means that the intermediate impedance Rq is real. This leads to two quadratic equations 



142 IMPEDANCE MATCHING 




MATCHING WITH TRANSMISSION LINES 



143 



A,. G 



of. = o>c, e = e. + e, 

' R, - R, 1 " 



»-#<'♦«'>-■ a>^ 

FIGURE 3.31 Transmission-line jt -transformer and relevant equations, 
to calculate the electrical lengths #i and 62 of the combined series transmission line: 



tan" 



tan 2 6? 



Z 0 Qi 



Z 0 Q 2 



'-< 1+ <*>(!)' 
.-0 + aS(|) : 



tanflj -1=0 



tan (9, -1=0 



(3.85) 



(3.86) 



where Qi = coCiRi and (22 = (oCoRi- 

However, to simplify the matching circuit design procedure, it is very helpful to use the nomographs 
shown in Figure 3.29. If the values of Ri and R 2 are selected in advance to set the intermediate 
resistance Ro, and the characteristic impedance of the transmission line Zo is known, the values of 61 
and 62 can be easily determined from one of these nomographs. 

A widely used two-port it -transformer with two shunt capacitors along with its design formulas is 
presented in Figure 3.31 [7,14]. Such a transformer can be conveniently used as an output matching 
circuit when the device collector or drain-source capacitance can be considered as a first shunt 
capacitance and parasitic series lead inductance can easily be added to a series transmission line. 
Also, it is convenient to use this transformer as the matching circuits in balanced power amplifiers, 
where the shunt capacitors can be connected between series transmission lines due to effect of virtual 
grounding. The schematic of a transmission-line two-port T-transformer with the series and shunt 
capacitors along with the design formulas is given in Figure 3.32. 



Ci 



o 



Z! o 

O 



mC, =— sin29, = 2Q,\^- - A. ] i»C, = -^Ai 



I + Q\ 



\R, cos ; 0, + (RJZ m )s\n 2 0, 
FIGURE 3.32 Transmission-line 7"-transformer and relevant equations. 



144 IMPEDANCE MATCHING 



3.4.2 Equivalence Between Circuits with Lumped and Distributed Parameters 

Generally, the input impedance of the transmission line at a particular frequency can be expressed as 
that of a lumped element, the equivalence of which for a shunt inductor L is shown in Figure 3.33(a) 
and for a shunt capacitor C is shown in Figure 3.33(b). If load represents a short when Z L = 0, from 
Eq. (3.63) it follows that 



Zm = ;'Z 0 tan( 



(3.87) 



which corresponds to the inductive input impedance for 6 < tz/2. The equivalent inductance at the 
design frequency on is calculated from 



_ ^in _ Z 0 tan 9 

OJ OJ 



(3.88) 



where X iT1 — ImZ in is the input reactance. This means that the network shunt inductor can equivalently 
be replaced with an open-circuited transmission line of characteristic impedance Z 0 and electrical 
length 6. 

Similarly, when Z L = oo, 



-y'ZocoK 



(3.89) 



which corresponds to the capacitive input impedance for ( 
design frequency oj is determined from 



< tt/2. The equivalent capacitance at the 



1 



ojX,, 



tan 6 

O)Z 0 



(3.90) 



These equivalences between lumped elements and transmission lines are exact only at the design 
frequency. The reactance of the inductor increases linearly with increasing frequency, while the 
reactance of a shorted line increases as tand. For a short transmission line when 9 90°, the input 
impedance increases linearly with frequency since tanf 0. Therefore, the short-circuit line behaves 
like an inductor and open-circuit line behaves like a capacitor over a range of frequencies where the 
electrical lengths of these transmission lines are much less than 90°. 



e<90° 



(</) 



i 



o 



Zo, 
8<90° 

-o 



(A) 

FIGURE 3.33 Equivalence between lumped element and transmission line. 




FIGURE 3.34 Lumped and transmission line matching-circuit equivalence. 



To define the single-frequency equivalence between n -transformers using lumped and distributed 
elements, it is convenient to use two-port transmission AfiCD-parameters. The transmission ABCD- 
matrices of these it -transformers shown in Figure 3.34(a) can respectively be given by 



[ABCD] L 



[ABCD] T 



1 0 

jo)C 1 

1 0 

jo)C 1 



1 jcoL 
0 1 



1 0 

jeoC 1 



1 — (o 2 LC jo)L 
jcoC (2 — co 2 LC^j 1 — co 2 LC 



cos 6 jZ 0 sin ( 

sin 6 

j cos 6 

Z 0 

cos0 — (oCjZo s'm8 



1 0 

jeoC 1 



j'Z 0 sin( 



— (2a>CjZo cos 9 + sin 8 — a/CjZ 2 sin 0) cosO — coCjZq sin ( 
Zo 



(3.91) 



(3.92) 



where 6 is the electrical length of a transmission line at the design frequency id. 

Since equivalent circuits must have equal matrix elements, then for A L — Aj and S L = Bj we can 
write 



1 — co 2 LC = cosO — (oCjZq sin 6* 
jcoL — jZo sin#. 



(3.93) 
(3.94) 



146 IMPEDANCE MATCHING 



As a result, the equivalent series inductance can be expressed through the parameters of the 
transmission line as 



Zo sin 6 



(3.95) 



whereas the relationship between the shunt capacitances C from a lumped 7r -transformer and the 
shunt capacitance Cj from a transmission-line it -transformer can be obtained by 



cosS + mCZns'mO — 1 

C T = . (3.96) 

coZq sin# 



From Eq. (3.96) it follows that Cj ^ C for high values of the characteristic impedance Z 0 when 
o)CZ 0 3> 1 or small values of the electrical length 8 when cosd 1. Consequently, the series lumped 
inductor L can be replaced by a short transmission line, the parameters of which can be calculated 
according to Eq. (3.95). The characteristic impedance of the series transmission line is chosen to be 
sufficiently high to provide better accuracy. 

Similarly, to define single-frequency equivalence between a lumped ^-transformer and a 
transmission-line transformer shown in Figure 3.34(6), their transmission ABCZ)-matrices can re- 
spectively be given by 



[ABCZ)] L = 



1 jwL 




1 0" 




1 ja)L 




0 1 




. J wC 1 . 




0 1 





1 - a) 2 LC jcoL (2 - a> 2 LC) ' 
jcoC 1 — ai 2 LC 



(3.97) 



[ABCD] T = 



1 7'ojLt 
0 1 



COS 0 

sin0 



j Zo sin t 
cos 8 



1 jajL-j; 
0 1 



wLj sin 8 ( coLj 

cosd 7Z0 I 2 cos£* + sinf 

Zo \ Z 0 



<d 2 L\ 
^0 



sin 8 



s'md 



cos8 



coLi sin0 



(3.98) 



where 8 is the electrical length of a transmission line at the design frequency co. 
For equivalent matrix elements Al = Ax and Cl = Or, 



1 - arLC = cos<9 ^sin0 (3.99) 

Zo 

sind 

ja)C = j — . (3.100) 
^0 



As a result, the equivalent shunt capacitance can be expressed through the parameters of the 
transmission line as 



sine' 

coZ 0 



(3.101) 



MATCHING WITH TRANSMISSION LINES 



147 



whereas the relationship between the series inductance L from a lumped T-transformer and the series 
inductance L T from a transmission-line transformer can be obtained by 

cos 9 H sinf? — 1 

L T = . (3. 102) 

— sin 6 
Z 0 

From Eq. (3. 102) it follows that L T s=s L for small values of the characteristic impedance Z 0 when 
coL/Z 0 3> 1 or small values of the electrical length 0 when cosS 1 . Consequently, the shunt lumped 
capacitor C can be replaced by a short transmission line, the parameters of which can be calculated 
according to Eq. (3.101). In this case, the characteristic impedance of the series transmission line is 
chosen to be sufficiently low to provide better accuracy. 



3.4.3 Narrowband Microwave Power Amplifier 

As an example, consider the design of a transmission-line output matching circuit for a 5 W 1 .6 GHz 
bipolar power amplifier that operates at a supply voltage of 24 V and provides a power gain of about 
10 dB. These requirements can be satisfied by using an n-p-n silicon microwave transistor intended 
to operate in class AB power amplifiers for a frequency range of 1.5 to 1.7 GHz. At the operating 
frequency of 1.6 GHz, let the output device impedance is Z 01It = (5.5 —76.5) Q, which corresponds 
to a series combination of the transistor output resistance and capacitance. To match this capacitive 
impedance to the standard 50 Q load, it is best to use a matching circuit in the form of a T-transformer 
shown in Figure 3.32. Figure 3.35 shows the complete two-port network including the output device 
impedance and matching circuit. 

The circuit should compensate for the series capacitance inherent in the output impedance. For 
the small electrical length when tan$i n i=s 9 m and characteristic impedance Zo S> Rout, Eqs. (3.79) and 
(3.80) in terms of the output resistance R onl and reactance X out can respectively be simplified to 



R™ 



R 2 



^ out 1 
Zo (wCoutZo 



(3.103) 
(3.104) 



where 62 is a part of the total transmission line that is required to compensate for the output capacitive 
reactance. If Z 0 is chosen to be 50 Q, then 0 2 — 6.5/50 = 0. 13 radians, which is equal to the electrical 
length of approximately 7.5°. Then, the value of quality factor Q 2 is defined by 



Q2 > 



1 = 2.84. 



/.<>■ 



1 

Ana 



-O- 



/,, 0, 



c, 



FIGURE 3.35 Complete output two-port network circuit. 



148 IMPEDANCE MATCHING 



Consequently, Q 2 must be larger than 2.84. For example, a value of Q 2 = 3 can be chosen to yield 
a 3-dB frequency bandwidth of 1.6 GHz/3 = 533 MHz. The values of the output matching circuit 
parameters are 



1 , 
B l = - sin" 1 
2 



C, 



c. 



2Q 



7(1 



Z 0 



= 21° 



2i = 



1 + Gl 



Ri cos 2 6»i + (R 2 /Z 0 ) 2 sin 2 < 



1 =0.5 



g 2 - Q 



coRi (1 + Gf) 
l 



4pF 



: 4pF. 



The function of each element for visual effect can be traced on the Smith chart, as shown in 
Figure 3.36. The easiest and most convenient way to plot the traces of the matching circuit elements 
is to first plot the traces of Qi and Q 2 , then plot the trace of the series transmission line as far as the 
intersection point with 22-circle, followed by the plot the trace of the shunt capacitance C 2 as far 
as the intersection point with Q l -circle, and finally plot the trace of the series capacitance Ci to the 




FIGURE 3.36 Smith chart with elements from Figure 3.35. 



MATCHING WITH TRANSMISSION LINES 



149 



center 50-Q point. The plot of the series transmission line represents an arc of the circle with the 
center point at the center of the Smith chart. 



3.4.4 Broadband UHF High-Power Amplifier 

Now consider the design example of a broadband 150 W power amplifier that operates over a 
frequency bandwidth of 470 to 860 MHz, uses a 28 V supply voltage, and provides a power gain 
of more than 10 dB. In this case, it is convenient to use a high power balanced LDMOS transistor 
specially designed for UHF TV transmitters as the active device. Let us assume that the manufacturer 
states the value of input impedance for each transistor-balanced part of Z in = (1.7 + 7 1 .3) Q at the 
center bandwidth frequency f c — V 470 • 860 = 635 MHz. The input impedance Z m represents a 
series combination of the input resistance and inductive reactance. To cover the required frequency 
bandwidth, low-<2 matching circuits should be used to reduce in-band amplitude ripple and improve 
input VSWR. 

To achieve a 3-dB frequency bandwidth, the value of a quality factor must be less than Q — 
635/(860 - 470) = 1.63. Based on this value of Q, the next step is to define a number of matching 
sections. For example, for a single-stage input lumped matching circuit, the value of quality factor Q is 



Q > 



50 
L7 



1 = 5.33 



which means that the entire frequency range can be appropriately cover using a multistage matching 
circuit only. 

In this case, it is important that the device input quality factor is smaller than 1.63, being equal to 
<2in — 1-3/1.7 = 0.76. It is very convenient to design the input matching circuit (as well as the output 
matching circuit) by using simple low-pass L-transformers composed of the series transmission line 
and shunt capacitance each, with a constant value of Q, for each balanced part of the active device. 
Then, these two input matching circuits are combined by inserting capacitances, the values of which 
are reduced twice, between the two series transmission lines. 

To match the series input inductive impedance to the standard 50-Q source, we use three low-pass 
L-transformers, as shown in Figure 3.37. In this case, the input resistance R in can be assumed to 
be constant over entire frequency range. At the center bandwidth frequency of 635 MHz, the input 
inductance is equal approximately to 0.3 nH. Taking this inductance into account, it is necessary 
to subtract the appropriate value of electrical length 0 in from the total electrical length # 3 . Due to 
the short size of this transmission line when tan6 m i=s Q iD , a value of 9 m can be easily calculated in 
accordance with 



<»L in 



(3.105) 



According to Eq. (3.83), there are two simple possibilities to provide input matching using a 
technique with equal quality factors of L-transformers. One option is to use the same values of 
characteristic impedance for all transmission lines; the other is to use the same electrical lengths for 



2a. 0: 



Zo}. 0) - 0, 




FIGURE 3.37 Complete broadband input two-port network circuit. 



150 IMPEDANCE MATCHING 



all transmission lines. Consider the first approach, which also allows direct use of Smith chart, and 
choose the value of the characteristic impedance Z 0 = Z 0 i = Z 02 = Z 03 — 50 f2. The ratio of input 
and output resistances can be written as 

R\ Ri R°* 

— = — = — (3.106) 

"2 «3 ^in 

which gives the values of R 2 — 16.2 Q and R^ — 5.25 Q for R S0UIce — Ri — 50 Q and R in — 1.7 Q. 
The values of electrical lengths are determined from the nomograph shown in Figure 3.29(a) as 6\ = 
30°, 6 2 = 7.5°, 0 3 = 2.4°. 

To calculate the quality factor Q, equal for each L-transformer, from Eq. (3.83), it is enough to 
know the electrical length 9\ of the first L-transformer. The remaining two electrical lengths can be 
directly obtained from Eq. (3.82). As a result, the quality factor of each L-transformer is equal to a 
value of Q — 1.2. The values of the shunt capacitances using Eq. (3.81) are C\ — 6 pF, C2 — 19 pF, 
C 3 = 57 pF. 

For a constant Q, we can simplify significantly the design of the matching circuit by using the 
Smith chart. After calculating the value of Q, it is necessary to plot a constant Q-circle on the Smith 
chart. Figure 3.38 shows the input matching circuit design using the Smith chart with a constant 
<2-circle, where the curves for the series transmission lines represent the arcs of the circles with center 




FIGURE 3.38 Smith chart with elements from Figure 3.37. 



MATCHING NETWORKS WITH MIXED LUMPED AND DISTRIBUTED ELEMENTS 



151 



point at the center of the Smith chart. The capacitive traces are moved along the circles with the 
increasing susceptances and constant conductances. 

Another approach assumes the same values of electrical lengths 0 — 61—62 — 63 and calculates 
the characteristic impedances of series transmission lines from Eq. (3.83) at equal ratios of the input 
and output resistances according to Eq. (3.106). Such an approach is more convenient in practical 
design, because, when using the transmission lines with standard characteristic impedance Zo = 50 £2, 
the electrical length of the transmission line adjacent to the active device input terminal is usually 
too short. In this case, it makes sense to set the characteristic impedance of the first transmission 
line to Z01 = 50 Q. Then, the value of 6 — 30° is determined directly from the nomograph shown in 
Figure 3.29(a). Further calculation of Q from Eq. (3.81) for fixed 6 and Z 0 /R 2 yields Q — 1.2. The 
characteristic impedances of the remaining two transmission lines are then calculated easily from 
Eq. (3.81) or Eq. (3.83). Their values are Z02 = 15.7 f2 and Z03 = 5.1 Q with the same values of the 
shunt capacitances. 



3.5 MATCHING NETWORKS WITH MIXED LUMPED AND 
DISTRIBUTED ELEMENTS 

The matching circuits, which incorporate mixed lumped and transmission line elements, are widely 
used both in hybrid and monolithic design technique. Such matching circuits are especially very con- 
venient when designing the push-pull power amplifiers where the shunt capacitances are simply 
connected between two series microstrip lines. In this case, the lumped matching circuits can be 
easily transformed to the transmission-line matching circuits based on the single-frequency equiva- 
lence between lumped and distributed elements. First, consider a periodic LC-structure in the form 
of the low-pass ladder it -network which is used as a basis for the lumped matching prototype. Then, 
the lumped prototype should be split up into individual 7r-type sections with equal capacitances by 
consecutive step-by-step process and replaced by their equivalent distributed network counterparts. 
Finally, the complete mixed matching structure is optimized to improve the overall performance by 
employing standard nonlinear optimization routine on the element values. For the single frequency 
equivalence between lumped and distributed elements, the low-pass lumped 7r-type ladder section 
with equal shunt capacitances can be made equivalent to a symmetrically loaded transmission line at 
the single frequency, as shown in Figure 3.39(a), where relationships between the circuit parameters 
are given by Eqs. (3.95) and (3.96). 

To provide the design method using a single frequency equivalent technique, the following con- 
secutive design steps can be performed [15]: 

• Designate the lumped 7r-type C\—Li—C 2 section to be replaced. 

• From the chosen 7r-type C\-L\-Ci section, form the symmetrical with equal capacitances C 
shown in Figure 3.39(£>), the choice of which is arbitrary, however the values cannot exceed the 
minimum of (Ci, C2). 

• Calculate the parameters of the equivalent symmetrical Cj-TL-Cj section using the parameters 
of the lumped equivalent it -section by setting the transmission-line electrical length 6 in accor- 
dance with Eqs. (3.95) and (3.96), assuming that the minimum of the capacitances Ci and C2 
should be greater or equal than Ct, so that Ct can be readily embedded in the new Cj-TL-Cj 
section. 

• Finally, replace the 7r-type C1-Z4-C2 ladder section by the equivalent symmetrical Cj-TL-Cj 
section and combine adjacent shunt capacitances, as shown in Figure 3.39(Z>), where the loaded 
parallel capacitances Ca and Cb are given as Ca = C[ + Ct and Cb — C' 2 + Cj. 

Figure 3.40(a) shows the electrical schematic of a broadband VHF high-power LDMOSFET 
amplifier. To provide an output power of about 15 W with a power gain of more than 10 dB in a 



152 IMPEDANCE MATCHING 



L 

YYYY 



=> C T d= 



L 



-o 

(a) 



o— 




I 

./-VY-V 



( ' 



■C 2 ' 
— O 



o- 



z„.e 



=FCi 



c T -r 



=rCi' = c'a-t 



/ . n 



=FC, 



FIGURE 3.39 Transforming design procedure for lumped and distributed matching circuits. 



frequency range of 225 to 400 MHz, an LDMOSFET device with the gate geometry of 1.25 vim x 
40 mm and a supply voltage of 28 V was chosen. In this case, the matching design technique is 
based on the multisection low-pass networks with the series transmission line and shunt capacitances, 
two 7r-type sections for input matching circuit, and one 7r-type section for output matching circuit. 
The sections adjacent to the device input and output terminals incorporate the corresponding internal 
input gate-source and output drain-source device capacitances. Since a difference between the device 
equivalent output resistance at the fundamental for several tens of watts of output power and the load 
resistance of 50 Q is not significant in this case, it is sufficient to be confined to only one section for 
output matching network. 

Once a matching network structure is chosen, based on the requirements to the electrical perfor- 
mance and frequency bandwidth, the simplest and fastest way is to apply an optimization procedure 
using CAD simulators to satisfy certain criteria over a wide frequency range. For such a broadband 
power amplifier, these criteria can be, for example, the minimum output power variation and input 
return loss with the maximum power gain and efficiency. To minimize the overall dimensions of the 
power amplifier board, the parallel microstrip line in the drain circuit can be treated as an element of 
the output matching circuit and its electrical length can be considered as a variable to be optimized as 
well. Applying a nonlinear broadband CAD optimization technique implemented in any high-level 
circuit simulator and setting the ranges of electrical length of the transmission lines between 0 and 
90° and parallel capacitances from 0 to 100 pF, we obtain the parameters of the input and output 
matching circuits. The characteristic impedances of all transmission lines can be set to 50 Q for 
simplicity and convenience of the circuit implementation. However, to speed up this procedure, it 
is best to optimize circuit parameters separately for input and output matching circuits with the 
device equivalent input and output impedances: a series ^C-circuit for the device input and a parallel 



REFERENCES 153 



28 V 



P m = I W O 




40.0 



225 250 275 300 325 350 375 J.MU/ 

(b) 

FIGURE 3.40 Circuit schematic and performance of broadband LDMOSFET power amplifier. 

i?C-circuit for the device output. It is sufficient to use a fast linear optimization process, which will 
take only a few minutes to complete the matching circuit design. Then, the resulting optimized values 
are incorporated into the overall power amplifier circuit for each element and final optimization is 
performed using a large-signal active device model. In this case, the optimization process is finalized 
by choosing the nominal level of input power with optimizing elements in much narrower ranges of 
their values of about 10-20% from nominal for most critical elements. For practical convenience all 
transmission lines might have the characteristic impedances of 50 Q. Figure 3.40(b) illustrates the 
simulated broadband power amplifier performance, where the output power is within the range of 
42.5 to 44.5 dBm with a power gain of 13.5 ± 1 dB in a frequency bandwidth of 225 to 400 MHz [7], 

REFERENCES 

1. S. Roberts, "Conjugate-Image Impedances," Proc. IRE, vol. 34, pp. 198-204, Apr. 1946. 

2. R H. Smith, "Transmission Line Calculator," Electronics, vol. 12, pp. 29-31, Ian. 1939. 

3. R H. Smith, Electronic Applications of the Smith Chart, New York: Noble Publishing, 2000. 



154 IMPEDANCE MATCHING 



4. P. S. Carter, "Charts for Transmission-Line Measurements and Computations," RCA Rev, vol. 3, pp. 355-368, 
Jan. 1939. 

5. H. A. Wheeler, "Reflection Charts Relating to Impedance Matching," IEEE Trans. Microwave Theory Tech, 
vol. MTT-32, pp. 1008-1021, Sept. 1984. 

6. W. L. Everitt, "Output Networks for Radio-Frequency Power Amplifiers," Proc. IRE, vol. 19, pp. 725-737, 
May. 1931. 

7. A. Grebennikov, RF and Microwave Power Amplifier Design, New York: McGraw-Hill, 2004. 

8. L. F. Gray and R. Graham, Radio Transmitters, New York: McGraw-Hill, 1961. 

9. V. M. Bogachev and V. V. Nikiforov, Transistor Power Amplifiers (in Russian), Moskva: Energiya, 1978. 

10. A. Tarn, "Network Building Blocks Balance Power Amp Parameters," Microwaves & RF, vol. 23, pp. 81-87, 
July. 1984. 

11. D. M. Pozar, Microwave Engineering, New York: John Wiley & Sons, 2004. 

12. Y. Ito, M. Mochizuki, M. Kohno, H. Masuno, T. Takagi, and Y. Mitsui, "A 5-10 GHz 15-W GaAs MESFET 
Amplifier with Flat Gain and Power Responses," IEEE Microwave and Guided Wave Lett, vol. 5, pp. 454—456, 
Dec. 1995. 

13. D. H. Steinbrecher, "An Interesting Impedance Matching Network," IEEE Trans. Microwave Theory Tech, 
vol. MTT-15, p. 382, June. 1967. 

14. A. V. Grebennikov, "Create Transmission-Line Matching Circuits for Power Amplifiers," Microwaves & RF, 
vol. 39, pp. 113-122, Oct. 2000. 

15. B. S. Yarman and A. Aksen, "An Integrated Design Tool to Construct Lossless Matching Networks with 
Mixed Lumped and Distributed Elements," IEEE Trans. Circuits and Systems - 1: Fundamental Theory Appl, 
vol. CAS-I-39, pp. 713-723, Sep. 1992. 



Power Transformers, Combiners, 
and Couplers 



It is critical, particularly at higher frequencies, that the special types of combiners and dividers 
are used to avoid insufficient power performance of the individual active devices. The methods of 
configuration of the combiners or dividers differ depending on the operating frequency, frequency 
bandwidth, output power, and size requirements. Coaxial cable combiners with ferrite cores are used 
to combine the output powers of power amplifiers intended for wideband applications. The device 
output impedance is usually sufficiently small for high power levels; so, to match this impedance with 
a standard 50-Q load, coaxial-line transformers with specified impedance transformation are used. 
For narrow-band applications, the ,/V-way Wilkinson combiners are widely used due to their simple 
practical realization. For microwaves, the size of combiners should be very small and, therefore, the 
hybrid microstrip combiners (including different types of the microwaves hybrids and directional 
couplers) are commonly used to combine output powers of power amplifiers or oscillators. In this 
chapter, the basic properties of three-port and four-port networks are presented, as well as a variety 
of different combiners, transformers, and directional couplers for application in RF and microwave 
transmitters. 



4.1 BASIC PROPERTIES 

Basic three-port or four-port networks can be used to divide the output power of a single power source 
or combine the output powers of two or more power amplifiers or oscillators. Generally, the multiport 
network required to combine the output powers of N identical power sources is based on these basic 
networks. In this case, it is very important to match the output impedances of all power amplifiers 
or oscillators with the load to provide the overall output power in N times larger than the output 
power of a single power amplifier or oscillator. Changes in the operation condition of one power 
amplifier or oscillator should not affect the operation conditions of the remaining power amplifiers 
or oscillators. To satisfy this requirement, all input ports of the combiner should be decoupled or 
mutually independent. When one of the power sources is eliminated, the total output power must 
decrease by as small as possible, being within the limits of a maximum permissible level. In addition, 
the combiners can effectively be used for both narrowband and broadband transmitters. 



4.1.1 Three-Port Networks 

The simplest devices used for power division and combining are the three-port networks with one 
input and two outputs in the power divider, as shown in Figure 4.1(a), and two inputs and one output 
in the power combiner, as shown in Figure. 4. 1(b). The scattering 5-matrix of an arbitrary three-port 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



155 



156 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



P m o- 




o P„ 



P,u2 



{a) 



> 








Power 


1 




combiner 

.1 






(*) 





-O A,„ 



FIGURE 4.1 Schematic diagrams of (a) power divider and (b) power combiner. 



network can be written by 



[S] 



Sn Sn S13 
S21 S22 S23 
S31 S32 S33 



(4.1) 



where S% — Sjj for the symmetric scattering ^-matrix when all components are passive and reciprocal. 
In this case, if all ports are appropriately matched (when Sa — 0), the scattering S-matrix is reduced to 



IS] = 



0 S n S 13 

512 0 S 23 

513 S 23 0 



(4.2) 



A lossless condition applied to a fully matched S-matrix given by Eq. (4.2) requires it to be a 
unitary matrix when 



1ST [S] = 1 



(4.3) 



where [S]* is the matrix, complex-conjugated to the original 5-matrix [1,2]. 
As a result of multiplying two matrices, 

|S 12 | 2 + |S 13 | 2 = |Si 2 | 2 + |5 23 | 2 = |5 13 | 2 + |S 23 | 2 = 
^nfe — S^Sn — S n Sn = 0. 



(4.4) 
(4.5) 



From Eq. (4.5) it follows that at least two of three available parameters 5i 2 , S13, and 5 23 should be 
zero, which is inconsistent with at least one condition given by Eq. (4.4). This means that a three-port 
network cannot be lossless, reciprocal, and matched at all ports. However, if any one of these three 
conditions is not fulfilled, a physically realizable device is possible for practical implementation. A 
lossless and reciprocal three-port network can be realized if only two of its ports are matched, or it is 
lossy being reciprocal and fully matched at all three ports, as in the case of the resistive divider. 

If a reciprocal three-port network represents the 3-dB power divider when, for a given input power 
at the port 1, the output powers at the ports 2 and 3 are equal, then, according to Eq. (4.4), 



IS12I = \S 13 \ 



1 



(4.6) 



4.1.2 Four-Port Networks 

The four-port networks are used for directional power coupling when, for a given input signal at 
the port 1, the output signals are delivered to the ports 2 and 3, and no power is delivered to the 
port 4 (ideal case), as shown in Figure 4.2. The scattering 5-matrix of a reciprocal four-port network 



BASIC PROPERTIES 157 



Input 



Pi o 



Isolate 



P 4 o- 



V 



Through 



-O IK 



Coupled 



-0/ J j 



FIGURE 4.2 Schematic diagram of directional coupler. 



matched at all its ports is given by 



IS] 



0 


Sr. 




Sl4 


Sn 


0 


s 23 


5 2 4 


Sl3 


Sa 


0 


S34 


Sl4 


S24 


S34 


0 



(4.7) 



where 5y = Sjj for the symmetric scattering 5-matrix when all components are passive and reciprocal. 
In this case, the power supplied to the input port 1 is coupled to the coupled port 3 with coupling 
factor |5b| 2 , whereas the remainder of the input power is delivered to the through port 2 with coupling 
factor |5 12 | 2 . 

For a lossless four-port network, the unitary condition of the fully matched 5-matrix given by Eq. 
(4.7) results in 



|S 12 | 2 + |5 13 | 2 



|S 12 | 2 + |S 24 | 2 = |Si 3 | 2 + |S 34 | 2 = |5 24 | 2 + |S 34 | 2 = 1 



(4.8) 



which implies a full isolation between ports 2 and 3 and ports 1 and 4, respectively, when 

S u = S 4l = S 23 = S 32 = 0 (4.9) 

and that 

|Si3l = l&il ISuI = ISmI. (4.10) 

The scattering 5-matrix of such a directional coupler, matched at all its ports with two decoupled 
two-port networks, reduces to 



IS] 



0 


S12 


5,3 


0 




0 


0 


S24 


S13 


0 


0 


S34 


0 


S24 


5 3 4 


0 



(4.11) 



The directional coupler can be classified according to the phase shift <p between two output ports 
2 and 3 as the in-phase coupler with <p — 0, quadrature coupler with <p — 90° or tt/2, and out-of- 
phase coupler with cf> — 1 80° or it . The following important quantities are used to characterize the 
directional coupler: 



• The power-split ratio or power division ratio K 2 , which is calculated as the ratio of powers at 
the output ports when all ports are nominally (reflectionless) terminated, 



158 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



• The insertion loss C12, which is calculated as the ratio of powers at the input port 1 relative the 
output port 2, 

C a = 101og 10 J = -201og 10 |S 12 | 

• The coupling C13, which is calculated as the ratio of powers at the input port 1 relative to the 
output port 3, 

P { 

C13 = lOlogio— = - 201og 10 |Si 3 | 

^3 

• The directivity C34, which is calculated as the ratio of powers at the output port 3 relative to the 
isolated port 4, 

C 34 = lOlogm^ = 201og 10 i-^j 

A 1^14 1 

• The isolation Cu and C23, which are calculated as the ratios of powers at the input port 1 relative 
to the isolated port 4 and between the two output ports (output port 2 is considered an input 
port), respectively, 

Pj 

C 14 = 101og 10 — = - 201og l0 |5 14 | 
P4 

Pi 

C 23 = 101og 10 — = - 201og 10 |5 23 | 



• The voltage standing wave ratio at each port or VSWR\, where i— 1, 2, 3, 4, calculated as 

1 + |Sul 

VSWR > = T^W\- 

In an ideal case, the directional coupler would have VSWRi — 1 at each ports, insertion loss 
C12 = 3 dB, coupling C13 = 3 dB, infinite isolation and directivity Cu = C23 = C34 = 00. 



4.2 TRANSMISSION-LINE TRANSFORMERS AND COMBINERS 

The transmission-line transformers and combiners can provide very wide operating bandwidths and 
operate up to frequencies of 3 GHz and higher [3,4]. They are widely used in matching networks 
for antennas and power amplifiers in the HF and VHF bands, in mixer circuits, and their low losses 
make them especially useful in high-power circuits [5,6]. Typical structures for transmission-line 
transformers consist of parallel wires, coaxial cables, or bifilar twisted wire pairs. In the latter case, 
the characteristic impedance can easily determined by the wire diameter, the insulation thickness, 
and, to some extent, the twisting pitch [7,8]. For coaxial cable transformers with correctly chosen 
characteristic impedance, the theoretical high-frequency bandwidth limit is reached when the cable 
length comes in order of a half wavelength, with the overall achievable bandwidth being about a 
decade. By introducing the low-loss high permeability ferrites alongside a good quality semirigid 
coaxial or symmetrical strip cable, the low frequency limit can be significantly improved providing 
bandwidths of several or more decades. 

The concept of a broadband impedance transformer consisted of a pair of interconnected trans- 
mission lines was first disclosed and described by Guanella [9,10]. Figure 4.3(a) shows a Guanella 



TRANSMISSION-LINE TRANSFORMERS AND COMBINERS 



159 




transformer system with transmission-line character achieved by an arrangement comprising one 
pair of cylindrical coils that are wound in the same sense and are spaced a certain distance apart by 
an intervening dielectric. In this case, one cylindrical coil is located inside the insulating cylinder 
and the other coil is located on the outside of this cylinder. For the currents flowing through both 
windings in opposite directions, the corresponding flux in the coil axis is negligibly small. However, 
for the currents flowing in the same direction through both coils, the latter may be assumed to be 
connected in parallel, and a coil pair represents a considerable inductance for such currents and acts 
like a choke coil. With terminal 4 being grounded, such a 1 : 1 transformer provides matching of the 
balanced source to unbalanced load and is called the balun (Z>a/anced-to-z(nbalanced transformers). 
In this case, if terminal 2 is grounded, it represents simply a delay line. In a particular case, when 
terminals 2 and 3 are grounded, the transformer performs as a phase inverter. 

A series-parallel connection of a plurality of coil pairs can produce a match between unequal source 
and load resistances. Figure 4.3(b) shows a 4: 1 impedance (2: 1 voltage) transmission-line transformer 
where the two pairs of cylindrical transmission line coils are connected in series at the input and in 
parallel at the output. For the characteristic impedance Zq of each transmission line, this results in 
the two times higher impedance 2Z 0 at the input and two times lower impedance Z 0 /2 at the output. 
By grounding terminal 4, such a 4: 1 impedance transformer provides impedance matching of the 
balanced source to the unbalanced load. In this case, when terminal 2 is grounded, it performs as a 4: 1 
unun (wnbalanced-to-imbalanced transformer). With a series-parallel connection of /; coil pairs with 
the characteristic impedance Z 0 each, the input impedance is equal to nZ 0 and the output impedance 
is equal to Zo/n. Since Guanella adds voltages that have equal delays through the transmission lines, 
such a technique results in the so called equal-delay transmission-line transformers. 

The simplest transmission-line is a quarterwave transmission line whose characteristic impedance 
is chosen to give the correct impedance transformation. However, this transformer provides a 



160 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



narrowband performance valid only around frequencies for which the transmission line is odd mul- 
tiples of a quarter wavelength. If a ferrite sleeve is added to the transmission line, common-mode 
currents flowing in both transmission line inner and outer conductors in phase and in the same 
direction are suppressed and the load may be balanced and floating above ground [11,12], If the 
characteristic impedance of the transmission line is equal to the terminating impedances, the trans- 
mission is inherently broadband. If not, there will be a dip in the response at the frequency at which 
the transmission line is a quarter-wavelength long. 

A coaxial cable transformer, the physical configuration, and equivalent circuit representation of 
which are shown in Figures 4.4(a) and 4.4(b), respectively, consists of the coaxial line arranged 
inside the ferrite core or wound around the ferrite core. Due to its practical configuration, the coaxial 
cable transformer takes a position between the lumped and distributed systems. Therefore, at lower 
frequencies its equivalent circuit represents a conventional low-frequency transformer shown in Fig- 
ure 4.4(c), while at higher frequency it is a transmission line with the characteristic impedance Z 0 
shown in Figure 4.4(d). The advantage of such a transformer is that the parasitic interturn capaci- 
tance determines its characteristic impedance, whereas in the conventional wire-wound transformer 
with discrete windings this parasitic capacitance negatively contributes to the transformer frequency 
performance. 

When R s — R L — Zq, the transmission line can be considered a transformer with a 1 : 1 impedance 
transformation. To avoid any resonant phenomena, especially for complex loads, which can contribute 
to the significant output power variations, as, a general rule, the length / of the transmission line is 
kept to no more than an eight of a wavelength X mi„ at the highest operating frequency, 



/ < 



(4.12) 



where is the minimum wavelength in the transmission line corresponding to the high operating 
frequency / max . 



Is 




FIGURE 4.4 Schematic configurations of coaxial cable transformer. 



TRANSMISSION-LINE TRANSFORMERS AND COMBINERS 



161 




(a) 



Outer 
conductor 




FIGURE 4.5 Low-frequency models of 1 : 1 coaxial cable transformer. 



The low-frequency bandwidth limit of a coaxial cable transformer is determined by the effect of 
the magnetizing inductance L m of the outer surface of the outer conductor according to the equivalent 
low-frequency transformer model shown in Figure 4.5(a), where the transmission line is represented 
by the ideal 1:1 transformer [6]. The resistance R 0 represents the losses of the transmission line. 
An approximation to the magnetizing inductance can be made by considering the outer surface of 
the coaxial cable to be the same as that of a straight wire (or linear conductor), which at higher 
frequencies where the skin effect causes the current to be concentrated on the outer surface, would 
have the self-inductance of 









L m = 21 




- 1 









nH 



(4.13) 



where I is the length of the coaxial cable in cm and r is the radius of the outer surface of the outer 
conductor in cm [6]. 

High permeability of core materials results in shorter transmission lines. If a toroid is used for the 
core, the magnetizing inductance L m is obtained by 



Ann (i — nH 



(4.14) 



where /; is the number of turns, fi is the core permeability, A e is the effective cross-sectional area of 
the core in cm 2 , and L e is the average magnetic path length in cm [13]. 

Considering the transformer equivalent circuit shown in Figure 4.4(a), the ratio between the 
power delivered to the load P L and power available at the source P$ — V^/SRs when R s — R L can 
be obtained from 



Pl 
Ps 



(2coL m ) z 
Rj + (2coL m ) 2 



(4.15) 



162 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



which gives the minimum operating frequency for a given magnetizing inductance L m , taking 
into account the maximum decrease of the output power by 3 dB, as 

/m,n>-^. (4.16) 

A similar low-frequency model for a coaxial cable transformer using twisted or parallel wires is 
shown in Figure 4.5(b) [6]. Here, the model is symmetrical as both conductors are exposed to any 
magnetic material and therefore contribute identically to the losses and low-frequency performance 
of the transformer. 

An approach using a transmission line based on a single bifilar wound coil to realize a broadband 
1:4 impedance transformation was introduced by Ruthroff [14,15]. In this case, by using a core mate- 
rial of sufficiently high permeability, the number of turns can be significantly reduced. Figure 4.6(a) 
shows the circuit schematic of an unbalanced-to-unbalanced 1:4 transmission-line transformer where 
terminal 4 is connected to the input terminal 1 . As a result, for V = V\ — V2, the output voltage is twice 
the input voltage, and the transformer has a 1:2 voltage step-up ratio. As the ratio of input voltage 
to input current is one-fourth the load voltage to load current, the transformer is fully matched for 
maximum power transfer when — 4R$, and the characteristic impedance of the transmission-line 
Z 0 is equal to the geometric mean of the source and load impedances: 

Z 0 = y/R^R[ (4.17) 

where R s is the source resistance and R L is the load resistance. Figure 4.6(b) shows an impedance 
transformer acting as a phase inverter, where the load resistance is included between terminals 1 and 
4 to become a 1 :4 balun. This technique is called the bootstrap effect, which does not have the same 
high-frequency response as Guanella equal-delay approach because it adds a delayed voltage to a 




FIGURE 4.6 Schematic configurations of Ruthroff 1 :4 impedance transformer. 



TRANSMISSION-LINE TRANSFORMERS AND COMBINERS 



163 




(b) 

FIGURE 4.7 Schematic configurations of 4: 1 coaxial cable transformer. 

direct one [16]. The delay becomes excessive when the transmission line reach a significant fraction 
of a wavelength. 

Figure 4.7(a) shows the physical implementation of the 4:1 impedance Ruthroff transformer 
using a coaxial cable arranged inside the ferrite core. At lower frequencies, such a transformer 
can be considered an ordinary 2:1 voltage autotransformer. To improve the performance at higher 
frequencies, it is necessary to add an additional phase-compensating line of the same length shown in 
Figure 4.7(Z>), resulting in a Guanella ferrite-based 4: 1 impedance transformer. In this case, a ferrite 
core is necessary only for the upper line because the outer conductor of the lower line is grounded at 
both ends, and no current is flowing through it. A current / driven into the inner conductor of the upper 
line produces a current / that flows in the outer conductor of the upper line, resulting in a current 21 
flowing into the load R L . Because the voltage 2V from the transformer input is divided in two equal 
parts between the coaxial line and the load, such a transformer provides impedance transformation 
from Rs — 2Zq into — Zq/2, where Zo is the characteristic impedance of each coaxial line. The 
bandwidth extension for the Ruthroff transformers can also be achieved by using transmission lines 
with step-function and exponential changes in their characteristic impedances [17,18]. To adopt this 
transmission-line transformer for microwave planar applications, the coaxial line can be replaced by 
a pair of stacked strip conductors or coupled microstrip lines [19,20]. 

Figure 4.8 shows similar arrangements for the 3:1 voltage coaxial cable transformers, which 
produce 9:1 impedance transformation. A current / driven into the inner conductor of the upper line 
in Figure 4.8(a) will cause a current / to flow in the outer conductor of the upper line. This current 
then produces a current / in the outer conductor of the lower line, resulting in a current 3/ flowing 
into the load R^. The lowest coaxial line can be removed, resulting in a 9:1 impedance coaxial 
cable transformer shown in Figure 4.8(£>). The characteristic impedance of each transmission line is 
specified by the voltage applied to the end of the line and the current flowing through the line and is 
equal to Zo. 

By using the transmission-line baluns with different integer-transformation ratios in certain con- 
nection, it is possible to obtain the fractional-ratio baluns and ununs [4,21,22]. Figure 4.9 shows a 
transformer configuration for obtaining an impedance ratio of 2.25 : 1 , which consists of a 1 : 1 Guanella 



164 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 




FIGURE 4.9 Schematic configuration of 



equal-delay 2.25:1 unun. 



TRANSMISSION-LINE TRANSFORMERS AND COMBINERS 



165 



balun on the top combined with a 1:4 Guanella balun where voltages on the left-hand side are in 
series and on the right-hand side are in parallel [21]. In this case, the left-hand side has the higher 
impedance. In a matched condition, this transformer should have a high frequency response similar 
to a single transmission line. By grounding the corresponding terminals (shown by dashed line), it 
becomes a broadband unun. Different ratios can be obtained with other configurations. For example, 
using a 1:9 Guanella balun below the 1:1 unit results in a 1.78:1 impedance ratio, whereas, with a 
1:16 balun, the impedance ratio becomes 1.56:1. 

On the other hand, the overall 1:1.5 voltage transformer configuration can be achieved by using 
the cascade connection of a 1:3 voltage transformer to increase the impedance by 9 times, and a 
2:1 voltage transformer to decrease the impedance by 4 times, which block schematic is shown in 
Figure 4.10(a) [22]. The practical configuration using coaxial cables and ferrite cores is shown in 
Figure 4.10(b). Here, the currents 1/3 in the inner conductors of two lower lines cause an overall 
current 2//3 in the outer conductor of the upper line, resulting in a current 2//3 flowing into the load 
i?L- A load voltage 3 V72 is out of phase with a longitudinal voltage V/2 along the upper line, resulting 
in a voltage V at the transformer input. The lowest line also can be eliminated with direct connection 
of the points at both ends of its inner conductor, as in the case of the 2:1 and 3:1 Ruthroff voltage 



21 




(b) 

FIGURE 4.10 Schematic configurations of fractional 1:2.25 impedance transformer. 



166 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



«o = 2Z< 
AAA^ 



o 




FIGURE 4.11 Coaxial cable combiner. 



transformers shown in Figures 4.7(a) and 4.8(b), respectively. If the source impedance is 50 Q, then 
the characteristic impedance of all three transmission lines should be 75 Q. In this case, the matched 
condition corresponds to a load impedance of 1 12.5 Q. 

By using the coaxial cable transformers, the output powers from two or more power sources can 
be combined. Figure 4.11 shows an example of such a transformer, combining two in-phase signals 
when both signal are delivered to the load R L and no signal will be dissipated in the ballast resistor 
Rq if their amplitudes are equal [14]. The main advantage of this transformer is the zero longitudinal 
voltage along the line for equal input powers; as a result, no losses occur in the ferrite core. When one 
input signal source (for example, power amplifier) defaults or disconnects, the longitudinal voltage 
becomes equal to half a voltage of another input source. For this transformer, it is possible to combine 
two out-of-phase signals when the ballast resistor is considered the load, and the load resistor in turn 
is considered the ballast resistor. 

The schematic of another hybrid coaxial cable transformer using as a combiner is shown in 
Figure 4.12. The advantage of this combiner is that both the load R L and the ballast resistor R 0 are 
grounded. These transformer-based combiners can also be used for the power division when the 
output power from single source is divided and delivered into two independent loads. In this case, 
the original load and the two signal sources should be switched. These hybrid transformer-based 
combiners can also be used for the power division when the output power from single source is 




/?o = 2Zo 

AM V\ Ks2 = Zo 



/ 



FIGURE 4.12 Two-cable hybrid combiner. 



TRANSMISSION-LINE TRANSFORMERS AND COMBINERS 



167 




(A) 

FIGURE 4.13 Coaxial cable combiners with increased isolation. 



divided and delivered into two independent loads. In this case, the original load and the two signal 
sources should be switched. As it turns out, the term "hybrid" comes not from the fact that the 
transformer might be constructed of two different entities (for example, cable and resistor), but just 
because it is being driven by two signals as opposed to only one. Consequently, the hybrid transformer 
represents a four-port device having two input ports, one sum port and one difference port. The unique 
characteristic of the hybrid transformer is ability to isolate the two input signal sources. 

Figure 4. 13(a) shows a coaxial cable two-way combiner, where the input signals having the same 
amplitudes and phases at ports 2 and 3 are matched at higher frequencies when all lines are of the 
same lengths and R & — Zq — Rl/2 — Rq/2 [4]. In this case, the isolation between these input ports can 
be calculated by 

C 23 (dB)= lOlog lo [4(l+4cot 2 0)] (4.18) 

where 6 is the electrical length of each transmission line. In order to improve the isolation, the 
symmetrical ballast resistor Ro should be connected through two additional lines, as shown in Figure 
4. 13(b), where all transmission lines have the same electrical lengths. 

Figure 4.14 shows a coaxial cable two-way combiner, which is fully matched and isolated in pairs 
[4]. Such combiners can be effectively used in high-power broadcasting VHF FM and VHF-UHF 



168 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 




FIGURE 4.14 Fully matched and isolated coaxial cable combiner. 



TV transmitters. In this case, for power amplifiers with the identical output impedances R sl and R$ 2 
when Rsi — R$2 — Zn/2, it is necessary to choose the values of the ballast resistor Rq and the load 
of R 0 — R L — Z 0 , where Z 0 is the characteristic impedance of the each transmission line of the same 
length. 



4.3 BALUNS 

Baluns are very important elements in the design of mixers, push-pull amplifiers, or oscillators to 
link a symmetrical (balanced) circuit to asymmetrical (unbalanced) circuit. Therefore, it makes sense 
to discuss their circuit configurations and performance in details separately. The main requirements 
to baluns are to provide an accurate 180-degree phase shift over required frequency bandwidth, with 
minimum loss and equal balanced impedances. In power amplifiers and oscillators, lack of symmetry 
will degrade output power and efficiency. Besides, the symmetrical port must be well isolated from 
ground to minimize an unwanted effect of parasitic capacitances. 

A wire- wound transformer whose simplified equivalent schematic is shown in Figure 4.15(a) 
provides an excellent broadband balun covering in commercial applications frequencies from low 
kHz to beyond 2GHz. They are usually realized with a center-tapped winding that provides a short 
circuit to even-mode (common-mode) signals while having no effect on the differential (odd-mode) 
signal. Wire-wound transformers are more expensive than the printed or lumped LC baluns, which 
are more suitable in practical mixer designs. However, unlike wire-wound transformers, the lumped 
LC baluns are narrowband as containing the resonant elements. 

Figure 4.15(b) shows the circuit schematic of a lattice-type LC balun that was proposed a long 
ago for combining powers in push-pull amplifier and their delivery to antenna [23]. It consists of two 
capacitors and two inductors, which produce the ±90-degree phase shifts at the output ports. The 
values of identical inductances L and capacitances C can be obtained by 



L = 



1 



(4.19) 
(4.20) 



where a> 0 is the center bandwidth frequency, R out is the balanced output resistance, and R L is the 
unbalanced load resistance. When designing this circuit, need to be confident that the operating 
frequency is well below the self-resonant frequencies of their components. 



BALUNS 169 




(c) 

FIGURE 4.15 Different circuit configurations of 1 : 1 balun. 

In monolithic microwave applications where the lumped inductances are usually replaced by 
transmission lines, the designs with microstrip coupled lines, Lange couplers, or multilayer cou- 
pled structures are very popular. However, the electrical length of the transmission lines at center 
bandwidth frequency is normally set to a quarter-wavelength, which is too large for applications in 
wireless communication systems. Therefore, it is very attractive to use the lumped-distributed balun 
structures, which can significantly reduce the balun size and, at the same time, can satisfy the required 
electrical characteristics. Figure 4.15(c) shows such a compact balun with lumped-distributed struc- 
ture consisting of the two coupled planar microstrip lines and two parallel capacitors, where the input 
transmission line is grounded at midpoint and the output transmission line is grounded at its one port 
[24]. Without these capacitors, it is necessary a very small spacing between quarterwave microstrip 
lines to achieve a 3-dB coupling between them. However, by optimizing the balun elements around 
the center bandwidth of 900 MHz, the planar structure of approximately one-sixteenth the size of the 
conventional quarter-wavelength structure was realized, with spacing 5=8 mils (note that 1 mil = 
0.0254 mm) using an FR4 board with substrate thickness of 300 mils. 

Figure 4.16 shows the circuit arrangement with two coaxial-line transformers combined to pro- 
vide a push-pull operation of the power amplifier by creating a balanced-to-unbalanced impedance 



170 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



Zo/2 

I — vw 






FIGURE 4.16 Circuit arrangement with two cable transformers for push-pull operation. 



transformation with higher spectral purity. Ideally, the out-of-phase RF signals from both active 
devices will have pure half-sinusoidal waveforms, which contain (according to the Fourier series 
expansion) only fundamental and even harmonic components. This implies a 180-degree shift be- 
tween fundamental components from both active devices and in-phase condition for remaining even 
harmonic components. In this case, the transformer T t representing a phase inverter is operated as 
a filter for even harmonics because currents flow through its inner and outer conductors in oppo- 
site directions. For each fundamental flowing through its inner and outer conductors in the same 
directions, it works as an RF choke, the impedance of which depends on the core permeability. 
Consequently, since the transformer T 2 represents a 1 : 1 balun, in order to provide maximum power 
delivery to the load R L , the output equivalent resistance of each active device should be two times 
smaller. 

For a simple 1:1 transmission-line balun realized with a twisted wire pair or coaxial cable, 
the balanced end is isolated from ground only at the center bandwidth frequency. To compensate 
for the short-circuited line reactance over certain frequency bandwidth around center frequency, a 
series open-circuited transmission line was introduced by Marchand, resulting in a compensated 
balun, the simplified schematic of which is shown in Figure 4.17(a) [25]. In this case, at center 
bandwidth frequency when the electrical length of the compensated line is a quarter-wavelength, 
the load resistance R L is seen unchanged. When this structure is realized with coaxial cables, to 
eliminate unwanted current existing in the outer conductor and corresponding radiation, it is neces- 
sary to additionally provide the certain coupling between the coaxial cables forming a transmission 
line with two outer conductors, as shown in Figure 4.17(b) [26]. Generally, the shunting reac- 
tance of this compensating line can reduce the overall balun reactance about center frequency or 
reverse it sign depending on the balanced load resistance, characteristic impedance of the com- 
pensating line, and coupling (characteristic impedance) between the outer conductors of two lines. 
Hence, a compensating line can create a complementary reactance to a balanced load and provide 
an improved match over broader frequency range. At microwaves, wire-wound transformers are 
usually replaced by a pair of the coupled transmission lines shown in Figure 4.17(c), thus result- 
ing in a compact planar structure. It should be noted that generally the characteristic impedances 
of the coaxial or coupled transmission lines can be different to optimize the frequency-bandwidth 
response. 

Multilayer configurations make Marchand balun even more compact and can provide wide band- 
widths due to the tight coupling between coupled-line sections. Modeling and synthesis result of a 
two-layer monolithic Marchand balun configuration with two-coupled lines, the basic structure of 
which is shown in Figure 4.18(a), is discussed in [27], In this configuration, the unbalanced terminal 
is connected to the microstrip line located at the upper metallization level, whereas the balanced load 
is connected to the microstrip lines located at the lower metallization level. The transmission-line 
sections in different layers are not isolated from each other. It should be noted that, for a given set 



BALUNS 



171 




XV4 
(M 

V4 XV4 



Inpul O 1 I 1 I O 




e > 



FIGURE 4.17 Schematic configurations of Marchand balun. 

of the output balanced and load unbalanced resistances R M and R L , the characteristic impedances of 
the outer and inner microstrip lines Z ol and Z 02 are not unique, and they can be calculated from 



Km O 




(4.21) 
(4.22) 



(*) 



FIGURE 4.18 Schematic configurations of coupled-line Marchand balun. 



172 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



where C is the coupling factor [28]. However, a different choice of Z 0 i and Z 02 leads to a different 
frequency bandwidth. For example, for R mt — 50 f2 and R L — 100 Q, it was found that using the 
symmetrical directional coupler with Zoi = Z02 = 40.825 Q results in a frequency bandwidth of 48.4% 
with ISiil < — 10 dB and amplitude imbalance within 0.91 dB, whereas the frequency bandwidth of 
20.9% with amplitude imbalance of less than 1.68 dB will be realized for the nonsymmetrical case 
when Zoi = 38 Q and Z 02 = 20.42 S2. 

The design of a three-line microstrip balun, the basic schematic of which is shown in Figure 
4.18(b), is based on the equivalence between a six-port section of three coupled lines and a six-port 
combination of two couplers [28]. The results of circuit analysis and optimization show that the 
spacings between adjacent microstrip lines are so narrow that it is difficult to fabricate a single-layer 
three-line balun. For a two-layer three-line balun with two coupled outer lines on the top metallization 
level, the spacing between these lines is significantly wider than in a single-layer case. However, wider 
frequency range can be achieved using a two-layer three-line balun with two coupled outer lines at 
the lower metallization level. For example, the measurements results for this balun show that, being 
fabricated on the Duroid RT5880 substrate, it can provide frequency range of 2.13 to 3.78 GHz with 
amplitude imbalance within 2.12 dB and phase error of less than 4.51°. 

To improve the performance of multilayer Marchand balun based on microstrip-line technology 
over frequency range, a short transmission line can be included connecting the two couplers, as shown 
in Figure 4. 19(a) [29]. This additional short microstrip line effectively compensates for the amplitude 
and phase imbalance caused by the difference in even- and odd-mode phase velocities. Besides, to 
minimize the balun size, the transmission lines of the coupler can be implemented in meander form 
that can give up to 90% reduction in size. As a result, the phase and amplitude differences of the 
compensated balun were within 180 ± 10° and 0 ± 1 dB over the frequency range of 5 to 30 GHz. The 
compensation can also be implemented by employing capacitors at each end of the coupled lines, as 




BALUNS 173 



shown in Figure 4.19(b) [30]. In this case, the capacitor will not affect the even-mode but effectively 
increases odd-mode phase length, thus resulting in a minimum amplitude and phase imbalance over 
certain frequency bandwidth. An exact synthesis technique that is widely used in filter design can 
be applied to develop and analyze new classes of miniaturized mixed lumped-distributed planar 
Marchand baluns using microstrip lines and lumped capacitors [31]. As an alternative, by employing 
two additional inductors at each balanced output and optimum coupling between the grounded strips 
shown in Figure 4. 19(c), a frequency bandwidth of 53% centered around 6.2 GHz with size reduction 
of 64% over a conventional coupled-line Marchand balun is achieved [32]. A combined compensation 
technique uses a series capacitor at the unbalanced input port to improve the matching bandwidth and 
inductors at the ground connections to minimize amplitude and phase imbalance [33]. 

Figure 4.20(a) shows the broadband parallel-connected coaxial cable balun as an alternative to a 
series-connected Marchand balun [34]. It consists of an unbalanced input coaxial cable connected to 
a dummy cable that maintains symmetry. On the opposite side of the balun, the output inner and outer 
conductors are connected in parallel to each other, while the input inner and outer conductors of coaxial 
cables are cross-connected. The right-hand portion of the balun forms a high impedance balanced 
load. By means of the cross connection, the high impedance is reduced to a low impedance showing 
a 4: 1 impedance transformation ratio, for example, from a balanced load of 200 Q to a single-ended 
50 Q. The frequency bandwidth of the balun is limited by the shunting effect at lower frequencies and 
near half-wave resonance. These parallel-connected baluns can provide approximately four times the 
operating frequency bandwidth of their series-connected counterparts as covering in the experiment 
the frequency range from 160 to 4000 MHz, a 25:1 bandwidth [35]. The parallel-connected balun 
may be realized in a variety of configurations, some of which are shown in Figures 4.20(b) and 
Figure 4.20(c) [34,35]. 




Input © 



Input O- 



1 



-vw- 



(c) 



FIGURE 4.20 Broadband parallel-connected cable 1:4 baluns. 



174 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



4.4 WILKINSON POWER DIVIDERS/COMBINERS 

The in-phase power combiners and dividers are the important components of the RF and microwave 
transmitters when it is necessary to deliver a high level of the output power to antenna, especially in 
phased-array systems. In this case, it is required to provide a high degree of isolation between output 
ports over some frequency range for identical in-phase signals with equal amplitudes. Figure 4.21(a) 
shows a planar structure of the basic parallel beam A-way divider/combiner, which provides a com- 
bination of powers from the A signal sources. Here, the input impedance of the N transmission lines 
(connected in parallel) with the characteristic impedance of Z 0 each is equal to Z 0 /N. Consequently, 
an additional quarterwave transmission line with the characteristic impedance Z 0 /\/A is required 
to convert the input impedance Z 0 /N to standard impedance Z 0 . However, this A'-way combiner 
cannot provide sufficient isolation between input ports. The impedances are matched only when all 
input signals have the same amplitudes and phases at any combiner input. The effect of any input 
on the remaining ones becomes smaller for combiners with greater number of inputs. For example, 
if the input signal is delivered into the input port 2 and all other (A — 1) input ports and output 
port 1 are matched, then the power dissipated at any load connected to the matched input ports will 
be decreased by (l — 1/A 2 ) /(2A — 1) times and isolation between any two input ports expressed 
through 5-parameters is obtained by 

/ 1 A 2 - 1 \ 

5,j = - 101og l(1 — (4.23) 

\N 2 2N — 1/ 

where A is a number of the input ports and i,j — 2,...,N+ 1 . 



Zo 




(h) 4> 
FIGURE 4.21 Circuit topologies of A'-way in-phase combiners/dividers. 



WILKINSON POWER DIVIDERS/COMBINERS 175 



In most cases, better isolation is required than obtained by Eq. (4.23). The simplest way to 
provide full isolation between the input and output ports of the combiner is to connect the ferrite 
isolators (circulators) at the input ports 2, . . . , N + 1. In this case, the lengths of the transmission lines 
connected between each ferrite isolator and a quarterwave transmission line should be equal. Although 
the ferrite isolators increase the overall weight and dimensions of the combiner and contribute to 
additional insertion losses, nevertheless they provide a very simple combiner realization and protect 
the connected power amplifiers from the load variations. By using such a 12-way parallel beam 
combiner, the continuous output power of 1 kW for the L-band transmitter was obtained at the 
operating frequency of 1.25 GHz [36]. 

When one or more power amplifiers are destroyed for some reasons, the overall output power P out 
and efficiency r/ c of the combiner can be calculated, respectively, by 

(N - M) 2 

Pout = - - Pi (4.24) 

P 0 ul M 

ric = — = 1 (4.25) 

Pin N 

where P in = (N — M)Pi, Pi is the output power from a single power amplifier, N is the number of 
the input ports, and M is the number of the destroyed power amplifiers. Part of the output power of 
the remaining power amplifiers will be dissipated within the ferrite isolators (in ballast resistors of 
circulators). For each ferrite isolator connected to the operating power amplifier, the dissipated power 
Pdo can be defined as 

/M\ 2 

Pdo= ( — J Pi (4.26) 

whereas for each isolator connected to the destroyed power amplifier, the dissipated power Pa can 
be calculated from 



N - M x 2 



N 



Pi- (4.27) 



In this case, by adding the ballast resistors R 0 — Z 0 , the right-hand side terminals of which are 
combined together in a common junction as shown in Figure 4.21(£>), matching of all ports, low loss 
and high isolation between input and output ports can be provided. Such kind of a simple N-way 
hybrid power divider is known as a Wilkinson power divider [37], However, it should be mentioned 
that, historically, this divider/combiner was reported a little bit earlier [3 8—40] . Originally, a Wilkinson 
power divider was composed of a coaxial line in which the hollow inner conductor has been split 
into N splines of length A/4, with shorting plate connecting the splines at the input end and resistors 
connected in a radial manner between each spline at the output end and a common junction. The 
frequency response of the voltage standing wave ratio at the divider input port, VSWR[ n , depending 
on the number of the output ports N, is shown in Figure 4.22 [41]. 

The hybrid planar microstrip realization of the simplest two-way Wilkinson divider is shown in 
Figure 4.23(a). It consists of the two quarterwave microstrip lines connected in parallel at the input 
end and the planar ballast resistor connected between the output ports of the microstrip lines. Despite 
its small dimensions and simple construction, such a divider provides a sufficient isolation between 
output ports over sufficiently wide frequency bandwidth when equal power division is provided due 
to a symmetrical configuration with ^ 0 — 2Z 0 and Zy — Z 0 ~j2. However, in practice, it is necessary 
to take into account the distributed RC structure of the ballast resistor when its size is sufficiently 
large, as well as manufacturing tolerances and discontinuities. As a result, in a frequency bandwidth 



176 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 




of 30% with VSWR in < 1.2 at input port 1 and VSWR 0M < 1.03 at output ports 2 and 3, the isolation 
between the divider outputs can be better than 20 dB [42], 

In millimeter-wave integrated circuits, in order to increase a self-resonant frequency of the ballast 
chip resistor, the overall MMIC dimensions must be very small. This means that the two branches of 
the power divider are very close to each other, which leads to strong mutual coupling between the 
output microstrip lines and, as a result, upsets the desired power-split ratio. A possible solution is to 
use the branches with the electrical lengths of 3A/4 instead of A/4 and to include the two additional 
branches into a semicircle, as shown in Figure 4.23(b) [43]. These additional branches should be of 
the half-wave electrical lengths with the characteristic impedances equal to Zq. In this case, isolation 
can be better than 17 dB between all ports with the insertion loss of about 1.3 dB at the operating 
frequency of 30.4 GHz. 




FIGURE 4.23 Microstrip realization of two-way Wilkinson dividers. 



WILKINSON POWER DIVIDERS/COMBINERS 177 



However, the ballast resistors of the conventional yV-way Wilkinson combiners/dividers cannot be 
designed to be a planar structure when their physical lengths and connecting wires are minimal to 
provide sufficient isolation among output ports over the required frequency range. For example, the 
radial and fork A'- way hybrids have reasonably wide frequency bandwidth, of about 20% and higher, 
but their match and isolation are not perfect even at the center bandwidth frequency [44]. Besides, due 
to small size of the ballast resistor compared to the wavelength and its balanced structure, it is difficult 
to heat-sink it in the case of high power combining. In order to provide higher output power capability, 
it is possible to modify the A'-way Wilkinson combiner/divider by replacing the ballast resistive star 
with a combination of quarterwave transmission lines and shunt-connected resistors [45], In this 
case, each ballast resistor is connected to corresponding output port through a transmission line. At 
the same time, all ballast resistors are connected to a common floating starpoint by the transmission 
lines. Such a modification has an advantage of external isolation loads (high-power ballast resistors) 
and easy monitoring capability for imbalances at the output ports. For a two-way planar power 
combiner/divider, the circuit topology of which is shown in Figure 4.24, the balanced 100-fi ballast 
resistor is replaced by a transmission-line network and two 50-Q resistors are connected to ground 
acting as the out-of-phase load, where Z] = Z 0 ^/2, Z 2 — Z 0 /*/2, and Z 0 = Rq — 50 Q [46]. 

The cascade connection of two-way Wilkinson power combiners/dividers can provide a mul- 
tiway power division or power combining. The simplest practical realization is the binary power 
divider/combiner, composed of the n stages when each consecutive stage of which contains an in- 
creasing by 2 N number of two-way dividers/combiners. For a single destroyed power amplifier, the 
power dissipated in the ballast resistors is equal to 




(4.28) 



The output power of P\I1 is dissipated in the ballast resistor adjacent to the destroyed power 
amplifier; the output power of PJ4 is dissipated in the ballast resistor of the next stage, and so on. 
It should be mentioned that the power divider with a number of outputs multiple to 4 N represents 
the convenient case when the characteristic impedance of the transmission line are of the same 
impedance. Figure 4.25 shows the four-way microstrip Wilkinson divider/combiner fabricated on 
alumina substrate with six 50-Q quarterwave microstrip lines and two 100 Q and one 50-Q thin-film 
resistors. This microstrip Wilkinson power divider/combiner can provide the insertion loss of less 



R 0 

A/V\/ 1|> 



— |l> 

j 

FIGURE 4.24 Gysel high-power in-phase planar combiner/divider. 




178 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



2 3 45 




I 

FIGURE 4.25 Practical four-way microstrip Wilkinson power combiner/divider. 



than 0.3 dB and isolation between any outputs of about 20 dB in a frequency bandwidth of ±10% in 
decimeter frequency band. 

The frequency bandwidth property of a Wilkinson power divider/combiner can be improved with 
an increasing number of its sections [42]. Generally, a broadband two-way Wilkinson power divider 
can contain N pairs of equal-length transmission lines and N bridging resistors distributed from input 
port 1 to output ports 2 and 3. For example, for N — 2, the theoretical minimum isolation in an octave 
band between ports 2 and 3 can achieve 27.3 dB with VSWR at each port better than 1 . 1 . In monolithic 
microwave integrated circuits (MMICs), by using a two-metal layer GaAs HBT process when the 
bottom metal layer can realize a coplanar waveguide (CPW) transmission line and the top metal layer 
can realize a microstrip transmission line, the size of a two-section two-way power divider/combiner 
can be reduced. In this case, an isolation of 15 dB and a return loss of 15 dB can be achieved in a 
frequency bandwidth from 15 to 45 GHz [47], 

Figure 4.26 shows the equivalent circuit representation of a three-way modified Wilkinson power 
divider/combiner [48]. Assuming all the impedances of the input and three output ports be 50 Q, 
the characteristic impedances of the quarterwave transmission lines are selected for a maximally flat 
performance as Zi = 1 14 Q and Z2 = 65.8 f2. To match circuit at the center frequency, the values 
of the ballast planar resistors should be chosen as ^1 = 64.95 Q and R 2 — 200 Q. In this case, the 
isolation between output ports of such a three-way divider demonstrates more than 20 dB in an octave 
frequency bandwidth. 

Generally, high characteristic impedance values (usually higher than 100 £2) for the transmission 
lines can create a problem in their practical microstrip implementation, since their narrow widths 
increase the insertion loss. In this case, using a recombinant power divider, the topology of which is 
shown in Figure 4.27, provides an isolation of 20 dB in a frequency range of 72% for a maximum 
line impedance of 80 Q and requires only three isolation resistors [49]. This three-way recombinant 



Input o- 



-O Output 
1 

-o Output 



R 3 



O Output 
3 



V4 X/4 
FIGURE 4.26 Microstrip three-way divider with improved isolation. 



WILKINSON POWER DIVIDERS/COMBINERS 179 



Input O — [ 




• Output I 



D O Output 2 



/ 



A' 



3 O Output 3 



FIGURE 4.27 Microstrip three-way recombinant divider with improved isolation. 



divider is characterized by the insertion loss of about 1 dB and return loss of more than 12 dB in 
a frequency range of 6 to 14 GHz, fabricated in 25-mil thick 99.6% alumina substrate. The design 
values for the quarterwave transmission lines were Z\ — 36 £2, Z 2 = Z3 = 40 £2, Z4 = 80 £2, and Z5 = 
Z 6 = 40 Q with the ballast resistors Ri = 50 Q and Ri— 100 £2, respectively. Over a 2: 1 bandwidth, 
the center-to-side and side-to-side isolations exceed 20 dB. 

The divider broadband properties can also be improved by using the more complicated phase- 
shifting circuit instead of a simple microstrip line. The phase shift between two output ports 2 and 
3 will be close to 90° in an octave frequency range if a Schiffman element based on the coupled 
microstrip lines is connected to one output port [50], At the same time, an additional microstrip line 
with the electrical length of 270° at the center bandwidth frequency should be connected to the second 
output port. 

In the design of a microwave distributed network, a power divider providing two equal-phase 
outputs with unequal power division is often required. The split-tee power divider is a simple compact 
and broadband device. It provides two isolated equal-phase unequal-amplitude outputs with a good 
match at each port. Since a split-tee power divider is similar to the N-way equiphase equiamplitude 
power divider, it can be developed from this A'-way divider as follows: connect M of the output ports 
together to form one port and the remaining (N — M) output ports together to form the other port, 
connect quarterwave transformers to the resulting output ports to adjust their impedance level, and a 
power divider with two equiphase outputs and power ratio of N/(N — M) is derived. 

The basic schematic of a power divider with unequal output load impedances is shown in Fig- 
ure 4.28(a) [51]. This power divider is designed so that, when fed from input port 1, the perfect 
match will be achieved at the center bandwidth frequency when the output power at port 3 is K 2 
times the output power at port 2, and the voltage between port 2 and ground is equal to the voltage 
between port 3 and ground when measured at equal distances from port 1. To satisfy these con- 
ditions, the characteristic impedances Zj and Z 2 for unequal loads R 2 — KZ 0 and R 3 — Z 0 /K are 
calculated from 



Z l = KZ 0 JK + 



K 



Z 0 / 1 

z 2 = —Jk + - 

K\ K 



(4.29) 
(4.30) 



where both transmission lines are of a quarter wavelength at the center bandwidth frequency. 

Since the voltages at port 2 and port 3 are equal with this design, a resistor may be placed between 
these two ports without causing any power dissipation. However, isolation between output ports and a 
good match seen looking in at any ports is obtainable because of this resistor. Finally, to transform the 
two unequal output impedances to output impedance Zq equal for each output port, the characteristic 



180 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



«i = Zo Zo (') 




Input 



Zo (1) 



Z, (2) « 2 = A'Z 0 , 

=■ — VW — |l' 



(a) 
(2) 



!□ vAA ||< 

(3) _ Z 0 

« 3 = K 

(4) Z 0 



>./4 



X/4 (•*) 
(*) 

FIGURE 4.28 Split-tee power divider. 



Outputs 



(5) Zo 



impedances of additional quarterwave transformers Z 3 and Z 4 and ballast resistor R 0 shown in Figure 
4.28(fc) are determined from 



Z3 = Zq\/~k 
z - Zo 

^0 = Z 0 \K + 



K 



(4.31) 
(4.32) 

(4.33) 



The three-way power divider with various output power ratios, which represents a planar structure 
and can be easily realized using microstrip lines with reasonable characteristic impedances, is shown 
in Figure 4.29 [52], When port 1 is an input port, the input power is divided by a ratio of M:N:K at 
corresponding output ports 2, 4, 6 with isolated ports 3 and 5 . The electrical lengths of the transmission 
lines must be 90° except for the half-wave middle horizontal line. The characteristic impedances of 
the transmission lines can be calculated from 











Zj = 


Zo 




(4.34) 




A 2 










z 2 = 


Zo, 




(4.35) 






M 


z 3 = 


Zo 




(4.36) 






[A2 




z 4 = 


Z °\ 


~N 


(4.37) 










z 5 = 


Zo, 


a: 


(4.38) 



where Ai = M + N + K and A 2 — N + K. For example, for a three-way divider with M — 3, 
N — 2 and A" = 1, it follows that Z\ — Z2 — 1.41Zo, Z4 = 1.22Zo and Z5 = 1.73Zq. The same 



WILKINSON POWER DIVIDERS/COMBINERS 181 




Z 4 (5) Z s 



(6) 



Zo 




FIGURE 4.29 New type of three-way power divider. 



characteristic impedances are required for a 1 : 1 : 1 equal-power three-way divider, only the input port 
must be changed to port 4 in this case. 

Figure 4.30 shows the compact microstrip three-way Wilkinson power divider designed to operate 
over a frequency range of 1.7 to 2.1 GHz, with minimum combining efficiency of 93.8%, maximum 
amplitude imbalance of 0.35 dB, and isolation better than 15 dB [53]. To avoid any amplitude and 
phase imbalances between the divider 50 Q output ports, the ballast resistor connected to its middle 
branch should be split into two equal parallel resistors. To obtain an ideal floating node, these two 




FIGURE 4.30 Compact microstrip three-way Wilkinson power divider. 



182 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



resistors are connected together with narrow microstrip lines that are as short as possible. Finally, to 
connect the resistors from both sides of the middle branch, a copper wire of 7-mil diameter is used. 
The most critical parameter is the isolation between port 2 and port 4, which can be improved by 
shortening the bondwire length. 



4.5 MICROWAVE HYBRIDS 

The branch-line couplers or hybrids were firstly described more than six decades ago; however, the 
problem of their exact synthesis remained a puzzle for a number of years [54]. Initially, the branch- 
line hybrid was analyzed as a four-arm symmetrical network based on a superposition of the results 
obtained in the even and odd modes [55]. By writing the even- and odd-mode matrices together, the 
characteristic impedances of the branch lines and coupling into different ports can be obtained. A 
general synthesis procedure that can be applied to any structure of a multibranch hybrid, based on an 
invariance of the Richard's variable S — jtanO to the transformation of S — > 1/5 apart from a 1 80° phase 
change, had become available a decade later [56], As a result, with highly precise computer-design 
techniques available for branch-line hybrids, it became possible to generate any coupling value in 
the useful 0-15-dB coupling range. Their waveguide designs that have been used in large complex 
feeds for phase-array radars are compact, highly predictable in amplitude and phase characteristics, 
and handle very high power. Coaxial, microstrip, or stripline implementations of branch-line hybrids 
provide simple planar structures of moderate bandwidth capability, up to about 2/3 of an octave. 

For a fully matched case with standard 50-Q source and load impedances, when the characteristic 
impedances of its transverse branches are 50 Q and the characteristic impedances of its longitudinal 
main lines are 50/V2 = 35.4 £2, the microstrip branch-line hybrid shown in Figure 4.31 represents 
a 3-dB directional coupler, for which power in arm 1 divides evenly between arms 2 and 3 with the 
phase shift of 90°. No power is delivered to arm 4, because the signal flowing through different paths 
(lengths of A./4 and 3A./4) have the same amplitude and opposite phases at this port. The branch-line 
hybrid does not depend on the load mismatch level for equal reflected coefficients from the outputs 
when all reflected power is dissipated in the 50-Q ballast resistor. However, in practice, due to the 
quarter-wavelength transmission-line requirement, the bandwidth of such a single-stage quadrature 
branch-line hybrid is limited to 10-20%. 

Figure 4.32 shows the calculated frequency bandwidth characteristics of a single-section branch- 
line hybrid matched at the center bandwidth frequency with the load impedance Zl = Zo = 50 £2 
[57]. At millimeter- wave frequencies, the lengths of the microstrip lines can actually get shorter than 
the widths and the mutual coupling between the input lines and discontinuities at the input increases 
significantly. This has a direct effect on the input/output match, frequency bandwidth, and isolation. 
To minimize the effect of these problems, the branch-line hybrid can be designed as a two-section 



Zo/V2 




Z0/V2 



Output 



Output 



FIGURE 4.31 Microstrip branch-line quadrature hybrid. 



MICROWAVE HYBRIDS 183 



C, : . C„ 























y 





























ill 



20 



0.7 0.8 0.9 1.0 I.I 1.2 f/fo 

FIGURE 4.32 Bandwidth performance of single-section branch-line hybrid. 

hybrid using three-quarterwave lines for the series main lines and quarterwave lines for the shunt 
branch lines, with all inputs/outputs orthogonal to each other [58]. As a result, the return loss can 
achieve 10 dB or better over 90% of the band, the isolation was 10 dB or better over the whole band, 
and the difference in the coupling can be equal or less than 1 dB over about 75% of the frequency 
band from 26 to 40 GHz. 

If one pair of terminating resistors has different values compared to the other pair, the resulting 
branch-line hybrid can operate as a directional coupler and an impedance transformer simultaneously 
[59]. Design values of the branch- and main-line characteristic impedances for a single-section 
branch-line hybrid shown in Figure 4.33, related to the input source impedance Z 0 s and output load 
impedance Zql, can be calculated by 



Z 2 = 



Z 3 = 



Zos 
~K 

Zos Zql 



1 + K 2 

Z{ Zol 

Zos 



(4.39) 
(4.40) 
(4.41) 



where K is the voltage-split ratio between output ports 2 and 3, and Rq — Zos [60]. Such a hybrid 
with a 2-to-l (50- to 25-Q) impedance transformation ratio can provide approximately 20-percent 



Input 



Isolated 



i||wwM*I 



XIA 

V4 )J4 
/ l 



Output 



Output 



FIGURE 4.33 Microstrip branch-line quadrature impedance-transforming hybrid. 



184 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



Input Z " s r 



Isolated 



i||wwM*ZZ[ 



/ 




Output 



Output 



Z, Z, 
FIGURE 4.34 Broadband microwave branch-line quadrature hybrid. 



frequency bandwidth with ±0.25-dB amplitude imbalance. However, for a fixed directivity, the 
frequency bandwidth of branch-line impedance-transforming hybrid increases as the output-to-input 
impedance ratio is reduced [59]. 

The operating bandwidth can be significantly increased using multistage impedance-transforming 
hybrids. A two-section branch-line impedance-transforming quadrature hybrid is shown in Fig- 
ure 4.34. To design this hybrid with the given impedance transformation ratio r and power-split ratio 
K 2 , the branch- and main-line characteristic impedances should be chosen according to 



Z\ = ZosJr 



Z\ _ / 



(?) 



Zi — Zr 



1-1 



(4.42) 
(4.43) 

(4.44) 



where t = rV 1 + K 2 [61]. The condition of Z2 = Z3 gives maximum bandwidth when the best 
performance at the center bandwidth frequency is specified. 

For an equal power division when K = 1, the condition of t = r*/2 specifies a minimum value of 
r, which is equal to 0.5. However, in practice, it is better to choose r in the range of 0.7 to 1 .3, in order 
to provide the physically realizable branch-line characteristic impedances for 50-Q. input impedance. 
For example, for the 50- to 35-fi impedance transformation using a two-stage hybrid, the impedances 
are as follows: Zi = 72.5 Q, Z 2 = Z 3 = 29.6 Q, and Z 4 = 191.25 Q. This gives the power balance 
between the output ports better than 0.5 dB with the return loss and isolation better than 20 dB over 
a frequency bandwidth of 25% for a 2-GHz hybrid. 

For MMIC applications, the overall size of the quadrature branch-line hybrids with quarterwave 
transmission lines is too large. Therefore, it is attractive to replace each quarterwave branch line 
with the combination of a short-length transmission line and two shunt capacitors providing the same 
bandwidth properties. Consider the admittance matrix [7J for a quarterwave transmission line shown 
in Figure 4.35(a) and the admittance matrix [Y b ] for a circuit consisting of a short transmission line 
with two shunt capacitors shown in Figure 4.35(b), which are given by 



[lb 



jZ 0 



0 -1 
-1 0 



1 



cos# — a>CZ sin(9 —1 
jZs'mO —1 cos 6 — coCZ sin t 



(4.45) 
(4.46) 



MICROWAVE HYBRIDS 185 



1 o 



(fl) 



z.e 



k/\2 



A 70.7 Q 



70.7 (i 



( ' 



X 



"r 0 



4 O- 



-O 2 



X/8 



X712 



-O 3 



<A> " (c) 

FIGURE 4.35 Reduced-size branch-line quadrature hybrid. 



where Z 0 is the characteristic impedance of a quarterwave line, Z and 8 are the characteristic 
impedance and electrical length of a shortened line, and C is the shunt capacitance. By equating the 
corresponding y-parameters in Eqs. (4.45) and (4.46), the simple ratios between the circuit parameters 
can be obtained in the form of 



C 



1 



CoZn COS ( 



(4.47) 
(4.48) 



from which it follows that the lengths of the hybrid transmission lines can be made much shorter by 
increasing their characteristic impedance Z. For example, when choosing the electrical length of 8 — 
45°, the characteristic impedance of the transmission line increases by a factor of a/2. 

A circuit schematic of the reduced-size branch-line quadrature hybrid is shown in Figure 4.35(c) 
[62]. Compared to the conventional branch-line hybrid with the characteristic impedances of its 
branch- and main-lines of Zo and Zq/^/2, respectively, the circuit parameters of the reduced-size 
branch-line hybrid are obtained from 



9 2 = sin 



/ Z 0 
UV2 



(4.49) 
(4.50) 

(4.51) 



where 8\ and 8 2 are the electrical lengths of the shunt branch-line and series main-line, respectively. For 
a particular case of the standard characteristic impedance Z 0 — 50 Q, the characteristic impedance and 
electrical lengths of the transmission lines are defined as Z = Zo\/2, 8\ = 45°, and #2 = 30°, as shown 
in Figure 4.35(c). Experimental results for a 25-GHz reduced-size branch-line quadrature hybrid show 
that its bandwidth performance is slightly narrower than that of the conventional quarterwave hybrid, 
but its overall size is more than 80% smaller. 



186 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



To further reduce the MMIC size, the transmission lines can be fully replaced by the lumped 
planar inductors. Such an approach becomes possible because the symmetrical lumped LC-type n- 
or T-section is equivalent at a single frequency to the transmission-line section with the appro- 
priate characteristic impedance and electrical length. The lumped-element equivalent circuit of a 
transmission-line branch-line hybrid is shown in Figure 4.36(a) [63]. This circuit has also some ad- 
ditional advantages when each its section can work as a separate impedance transformer, a low-pass 
filter, and a phase shifter. The circuit can be diced into four separate sections and cascaded for the 



I O- 



4 O- 





I 








1 = 


-C c - 


























1 





































-O 2 



-0 3 



(a) 




4 O 



O 2 1 O- 



O 3 4 O- 



-O 2 



-O 3 



(b) 



(c) 




O 2 I O 



O 3 4O 




O 2 



O 1 



FIGURE 4.36 Equivalent circuits of lumped LC-type hybrid. 



MICROWAVE HYBRIDS 187 



desired transmission characteristics. The circuit analysis indicates that various types of networks 
fulfill the conditions required for an ideal hybrid. Therefore, greater design flexibility in the choice 
of the hybrid structure and performance is possible. 

Several possible single-section two-branch hybrid options are shown in Figure 4.36 [64]. In 
this case, it should be noted that not only a low-pass section but also a high-pass section can be 
respectively used. In the latter case, the high-pass LC section is considered an equivalent single- 
frequency replacement for a 270-degree transmission line [65]. Figures 4.36(c) and 4.36(d) illustrate 
the use of both low-pass and high-pass sections simultaneously, while only high-pass sections compose 
the hybrid shown in Figure 4.36(e). The performances of the hybrids shown in Figures 4.36(b) and 
4.36(e) are very similar to that of the classical single-section branch-line hybrid. The bandwidth 
performances of the hybrids shown in Figures 4.36(e) and 4.36(d) are narrower because the power 
balance between their output ports is much narrower. Broader bandwidth and lower output impedances 
can be provided with a two-section three-branch lumped-element hybrid. 

Figure 4.37(a) shows the equivalent circuit of a capacitively coupled lumped-element hybrid, 
which is used for monolithic design of variable phase shifters [66]. However, the power and phase 
balance bandwidth at the output ports of this hybrid is very narrow, in the limits of a few percent. An 
alternative design of an inductively coupled lumped-element hybrid is shown in Figure 4.37(b) [67], 
As a basic element, it includes lumped multiturn mutually coupled spiral inductors with the coupling 
coefficient, which can be realized using a bifilar (a sandwich of two multiturn spiral inductors with 
inner and outer windings) spiral transformer to achieve a coupling coefficient k — 0.707. In this 
case, this basic lumped-element configuration is completely equivalent to a transmission line in the 
vicinity of the center bandwidth frequency. The inductively coupled hybrid can provide a power 
balance within 0.2 dB and phase balance within 1° in a frequency bandwidth of ±10% in 2-GHz 
wireless applications. However, during the design procedure, some parasitic effects should be taken 



1 O 




O 2 



4 O 



O 3 




(a) 




1 O 



■O 2 




FIGURE 4.37 Equivalent circuits of lumped hybrid with capacitive and inductive coupling. 



188 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



into account. For example, the coupled inductor itself has a sufficient value of internal capacitance. 
Also, the finite value of the inductor quality factor results in a modest amplitude imbalance, but 
it leads to a significant phase deviation from ideal quadrature 90-degree difference. In this case, to 
compensate for the resulting performance degradation, the electromagnetic simulation of the structure 
and optimization of the values of the added shunt capacitors on both sides of the circuit are required. 
Two of these inductively coupled hybrids can be cascaded in tandem to significantly extend the 
frequency bandwidth. As a result, the phase shift of 93 ± 6°, the insertion loss between 1 and 1.5 dB, 
the return loss better than 16 dB, and the isolation between the output ports better than 18 dB were 
measured over the frequency range from 2 to 6 GHz [67]. 

A hybrid-ring directional coupler or rat-race, which is one of the fundamental components used in 
microwave circuits was described and analyzed more than six decades ago [68]. Its operation principle 
is based on an assumption that the voltage at any point along the transmission line is a superposition 
of the forward and backward propagating waves. Signal from the excitation source spreads out in the 
driving point and propagates along the line. As the forward wave reaches the far-end termination, it 
reflects, propagates backward, reflects from the near-end termination, propagates forward again, and 
continues in a loop. According to this wave behavior, a pure standing wave is set up within the ring 
when there is no mechanism for dissipation other than the minor ohmic losses associated with wave 
transmission. The point of voltage minimum (zero) corresponds to the case when the waves have 
phase shift of 180° with respect to each other. As a result, the standing wave within the ring can be 
mapped by marking off alternate voltage maxima and minima at quarterwave intervals. Consequently, 
if standing waves are set up in the main arm, the side arm receives maximum power when a voltage 
minimum of the standing-wave pattern coincides with the center of this arm and minimum power 
when a voltage maximum located in this point. 

The 3 A/2 ring hybrid, whose microstrip topology is shown in Figure 4.38(a), can be used to divide 
the driving power or to combine the powers from two sources. In the case of power division, a signal 
applied to the port 1 will be evenly split into two in-phase components at ports 2 and 3, and port 4 will 
be isolated. If the input signal is applied at port 4, it will be equally split into two components with 
180-degree difference at ports 2 and 3, and port 1 will be isolated. When operated as a combiner, with 
input signals applied at ports 2 and 3, the sum of the input signals will be delivered to port 1. Their 
difference will flow to port 4 and dissipated in corresponding ballast resistor Rq . For equal signals and 
matched ring hybrid with Z 0 = Rq, there is zero-power dissipation at port 4 [55]. The ideal rat-race 
or ring hybrid has the bandwidth of approximately 27.6% at the tolerance limits of the deviation of 
0.43 dB for split and of 20 dB for the maximum return loss and isolation. 

Generally, a hybrid-ring directional coupler can provide arbitrary power divisions when the power- 
split ratio is adjusted by varying the impedances between the arms [69]. Figure 4.38(b) shows the 
planar microstrip topology of a ring hybrid where the characteristic impedances of four arms are 
equal to the standard characteristic impedance Z 0 . The variable parameters are the two characteristic 
impedances Z\ and Z%, which determine the degree of coupling of the output arms and the impedance 
matching condition for the input arm. The analysis of this ring hybrid consists of the usual procedure of 
reducing the four-terminal network to a two-terminal network by taking advantage of the symmetry 
about the plane A-A' . For example, when two in-phase waves of equal amplitude are applied to 
terminals 1 and 3, the current is zero at the plane A-A' . As a result, the ring can be open-circuited at 
this plane and only one-half of the circuit can be analyzed. This condition is called the even mode. 
On the other hand, the odd-mode condition is a result of applying the opposite-phase waves of equal 
amplitude to terminals 1 and 3 when the voltage at plane A-A' is zero. In this case, the ring can be 
short-circuited at this plane and only one half of the circuit can be analyzed. Once the scattering 
matrices for the even and odd modes are known, the reflected waves in each arm can be determined. 
Then, by superimposing the waves of the two modes, the resultant reflected waves in each arm and a 
single incident wave in one arm is obtained. Thus, if arm 1 is input, the output voltage ratio between 
arm 2 and 3 is equal to Z1/Z2, and no power is delivered to the isolated port 4 where the ballast resistor 
R 0 — Z 0 is connected. 



MICROWAVE HYBRIDS 189 



1 3 




3X/4 

I <*) 



A' 

FIGURE 4.38 Microstrip rate-race ring hybrids. 

Although usually considered a narrowband device, the rat-race hybrid can provide much broader 
performance if its three-quarter wavelength section (a main limiting factor in a conventional config- 
uration) is replaced by one having the same characteristic impedance, but whose electrical length 
is realized by a quarter wavelength of line exhibiting the characteristics of an ideal phase-reversing 
network [70]. It may be a coaxial cable with a crossover of its inner and outer conductors at one 
end, thus resulting in an isolation of more than 20 dB in a frequency bandwidth of 30% [57]. As 
a compact planar alternative, a pair of equilateral broadside-coupled segments of strip transmission 
lines having diametrically opposing ends short-circuited, as shown in Figure 4.39(a), approximates a 
phase-reversing network over a wide frequency range. 

The characteristic impedance of the coupled section Z c is given by 



2Z 0e Zoo sin£ 



yf (Z 0e - Z 0o ) 2 - (Z 0e + Z 0o ) 2 COS 2 ( 



(4.52) 



190 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 





3X74 



FIGURE 4.39 Microstrip broadband ring hybrids. 



where Z 0e is the even-mode impedance, Z 0o is the odd-mode impedance, and 6 is the electrical length 
of the coupled region [71]. To realize a required 270° of phase shift, 6 must be 90°, resulting in 



2Z(vZn, 



(4.53) 



Z()e — Zo 0 

which for proper operation should be equal at midband to Zo\/2. Since Z c = V Zo e Zo 0 , 

Z 0e = (2 + V2~) Z 0 (4.54) 



(2 - V2) Z 0 



(4.55) 



MICROWAVE HYBRIDS 191 



which means that, for a 50-fi transmission system and a median 3-dB coupling, Z 0e must be 170.7 Q 
and Z 0o must be 29.3 Q. 

To extend the frequency bandwidth and simplify fabrication process with a uniplanar structure, 
the ring hybrid can represent a slotline ring with one slotline and three CPW arms [72]. The design 
technique substitutes one reverse-phase slotline T-junction for the conventional rat-race phase delay 
section. Since the phase reverse of the slotline T-junction is frequency independent, the resulting slot- 
line ring hybrid provides a broad bandwidth. Experimental results show that this uniplanar crossover 
ring-hybrid coupler has a bandwidth from 2 to 4 GHz with ±0.4 dB power dividing balance and ±1° 
phase balance. By using a microcoplanar stripline ring with a broadband coplanar stripline phase 
inverter and four CPW arms, the coupling between the output ports is within 3.5 ± 0.5 dB over the 
frequency range from 1.43 to 2.95 GHz [73]. 

However, it is possible to increase the operating bandwidth of the conventional rat-race without 
a quarterwave phase-reversing section by applying the concept of hypothetical ports and adding the 
proper quarterwave transformer to each port [74]. The bandwidth will be significantly broadened by 
replacing a three-quarterwave equal-impedance section with symmetrical transmission-line structure, 
the middle quarterwave section of which has different characteristic impedance. To optimally realize 
this, the numerical optimization procedure to minimize an objective evaluation function is required. 
In this case, to improve coupling, matching, and isolation, it is necessary to connect either the open- 
circuited compensating stubs at the hypothetical ports (connecting points of the quarterwave lines 
with different characteristic impedances), or to add the quarterwave transformers (transmission lines 
with different characteristic impedances) to the hybrid ports, as shown in Figure 4.39(i). As a result, 
the frequency bandwidth of the improved ring hybrid becomes 1.84 times as wide as the conventional 
rat-race. 

To significantly reduce the overall size of the conventional ring hybrid with the 90-degree and 
270-degree transmission line sections, which is crucial for MMIC design, the principle of the single- 
frequency equivalence between the circuits with the lumped and distributed parameters can be applied 
[65]. Let us compare the elements of the transmission matrix [ABCD^] for a 270-degree transmission 
line and the transmission matrix [ABCD b ] for a high-pass 7r-type lumped circuit, consisting of the 
series capacitor and two shunt inductors, given by 



[ABCDJ 



[ABCD b ] 



cos 6 j Zo sin ( 

sin 8 

i cos 0 

Zo 



1 0 

-j/coL 1 



S=3ir/2 

-j/coC 
1 



0 



L Zo 



1 



-jZ 0 
0 



0 



-j/toL 1 



1 



uP-LC 



^ coL ' 



1 



o 2 LC 



(4.56) 

1 " 

J—p; 
coC 

1 

a> 2 LC. 
(4.57) 



Equating the corresponding elements of both matrices yields the simple equations to determine 
the circuit elements in the form of 



ZqwC 



Zo_ 
coL 



(4.58) 



Hence, to reduce the size of a ring hybrid, a 270-degree transmission line shown in Figure 4.40(a) 
is replaced by a high-pass lumped section. Then, the three quarterwave-line sections are replaced by 
shortened ones with two shunt capacitors, as shown in Figure 4.35(fe) for a branch-line coupler. Finally, 
the characteristic impedance of the transmission line sections is chosen so that the shunt inductors 
of the high-pass section shown in Figure 4.40(£>) could resonant with the shunt capacitors of the 



192 POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



Zo.31/4 




(b) (c) 

FIGURE 4.40 Reduced-size ring hybrid. 

low-pass section at the center bandwidth frequency, in order to completely remove these components. 
Figure 4.40(c) shows the circuit diagram of the reduced-size ring hybrid with the characteristic 
impedances of the transmission-line sections of 100 £2 and their electrical lengths of 45° [62]. As 
a result, the overall reduced hybrid size is more than 80% smaller than that of the conventional 
hybrid. 

4.6 COUPLED-LINE DIRECTIONAL COUPLERS 

The first directional couplers consisted of either a two- wire balanced line coupled to a second balanced 
line along a distance of quarter wavelength, or a pair of rods a quarter wavelength long between ground 
planes [54]. Although the propagation of waves on systems of parallel conductors was investigated 
many decades ago in connection with the problem of crosstalk between open wire lines or cable pairs 
in order to eliminate the natural coupling rather than use it, the first exact design theory for transverse 
electromagnetic (TEM) transmission-line couplers was introduced by Oliver [75]. In terms of the 
even and odd electric-field modes describing a system of the coupled conductors, it can be stated 
that the coupling is backward with coupled wave on the secondary line propagating in the direction 
opposite to the direction of the wave on the primary line, the directivity will be perfect with VSWR 
equal to unity if Z\ — Zo e Z 0o at all cross sections along the directional coupler, and the midband 
voltage coupling coefficient C of the directional coupler is defined as 

Zne — Znn 

C = — (4.59) 

Zoe + Zo 0 

where C — 0 for zero coupling and C — 1 for completely superposed transmission lines. 

A coupled-line directional coupler, the stripline single-section topology of which is shown in 
Figure 4.41(a), can be used for broadband power division or combining. Its electrical properties are 
described using a concept of two types of excitations for the coupled lines in TEM approximation. 
In this case, for the even mode, the currents flowing in the strip conductors are equal in amplitude 
and flow in the same direction. The electric field has even symmetry about the center line, and no 
current flows between the two strip conductors. For the odd mode, the currents flowing in the strip 
conductors are equal in amplitude, but flow in opposite directions. The electric field lines have an 
odd symmetry about the center line, and a voltage null exists between these two strip conductors. 



COUPLED-LINE DIRECTIONAL COUPLERS 193 



Through 



Input 



Isolated Isolated 
— ^/v |l< 'l| W\< — 



4 *o 



Through 



/. 4 



Coupled Input 



Coupled 




Through 



FIGURE 4.41 Coupled-line directional couplers. 



An arbitrary excitation of the coupled lines can always be treated as a superposition of appropriate 
amplitudes of even and odd modes. Therefore, the characteristic impedance for even excitation mode 
Zoe and the characteristic impedance for the odd excitation mode Zq 0 characterize the coupled lines. 
When the two coupled equal-strip lines are used in a standard system with characteristic impedance 
of Z 0 , Z\ = Z 0e Z 0o and 



n + c 

Zoe = Zo VT^c (4 ' 60) 
[T~c 

Zoo = z °VTTc' (4 ' 61) 



An analysis in terms of scattering 5-parameters gives Sn = S14 = 0 for any electrical lengths 
of the coupled lines and the output port 4 is isolated from the matched input port 1. Changing the 



194 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



coupling between the lines and their widths can change the characteristic impedances Z 0e and Z 0o . In 
this case, 

vT^C 2 

S12 = ; (4.62) 

Vl - C 2 cosS + j sine 

jC sin 9 

Sn = . (4.63) 

Vl - C 2 cos6» + j sine 

where 6 is the electrical length of the coupled-line section. 

The voltage-split ratio K is defined as the ratio between voltages at port 2 and port 3 by 



K 



Sn 
Sn 



Vi-c 2 

C sine 



(4.64) 



where K can be controlled by changing the coupling coefficient C and electrical length e. 
For a quarter-wavelength-long coupler when e — 90°, Eqs. (4.62) and (4.63) reduce to 

Sn = ~ jJl-C 2 (4.65) 
S u = C (4.66) 

from which it follows that equal voltage split between the output ports 2 and 3 can be provided with 
C = 1/V2. 

If it is necessary to provide the output ports 2 and 3 at one side, it is best to use a construction 
of a microstrip directional coupler with crossed bondwires, as shown in Figure 4.41(b). The strip 
crossover for a stripline directional coupler can be easily achieved with the three-layer sandwich. The 
microstrip 3-dB directional coupler fabricated on alumina substrate for idealized zero strip thickness 
should have the calculated strip spacing of less than 10 p.m. Such a narrow value easily explains 
the great interest to the constructions of the directional couplers with larger spacing. The effective 
solution is to use a tandem connection of the two identical directional couplers, which alleviates 
the physical problem of tight coupling, since two individual couplers need only 8.34-dB coupling 
to achieve a 3-dB coupler [76,77]. The tandem coupler shown in Figure 4.41(c) has the electrical 
properties of the individual coupler when the output ports 1, 4 and 2, 3 are isolated in pairs, and the 
phase difference between the output ports 2 and 3 is of 90°. 

From an analysis of the signal propagation from input port 1 to output ports 2 and 3 of the 
tandem coupler, when the signal from the input port 1 propagates to the output port 2 through the 
traces l-2'-l'-2 and l-3'-4'-2 while the signal flowing through the traces l-2'-l'-3 and l-3'-4'-3 is 
delivered to the output port 3, the ratio of the scattering parameters Sj? and Sj 3 of a tandem coupler 
can be expressed through the corresponding scattering parameters Sn and Sn of the individual 
coupler as 

S T n = Sh + Sh = . l-C 2 (l+sm 2 e) 
Sj 3 2S l2 Sn 2CVl - C 2 sin0 

As a result, the signal at the port 2 overtakes the signal at the port 3 by 90°. In this case, for a 3-dB 
tandem coupler with 9 — 90°, the magnitude of Eq. (4.67) must be equal to unity. Consequently, the 
required voltage coupling coefficient is calculated as 



C = 0.5V 2- V2 = 0.3827 



Isolated Z 




X/4 



Input 



COUPLED-LINE DIRECTIONAL COUPLERS 195 



Through 



/. 4 



Coupled 



Input 



■VW |l< 



Z-4. Z 



Zo Zo 
(a) 



7V 



Coupled 



Zo 



(!>) 



A, 



FIGURE 4.42 Lange directional couplers. 



c, 



Cr 



8.34 dB. 



As an example, a tandem 8.34-dB directional coupler has the dimensions of W/h — 0.77 and 
S/h = 0.18 for alumina substrate with e r = 9.6, where W is the strip width, 5 is the strip spacing, and 
h is the substrate thickness [57]. 

Another way to increase the coupling between the two edge-coupled microstrip lines is to use 
several parallel narrow microstrip lines interconnected with each other by the bondwires, as shown 
in Figure 4.42. For a Lange coupler shown in Figure 4.42(a), four coupled microstrip lines are used, 
achieving a 3-dB coupling over an octave or more bandwidth [78]. In this case, the signal flowing 
to the input port 1 is distributed between the output ports 2 and 3 with the phase difference of 90°. 
However, this structure is quite complicated for practical implementation when, for alumina substrate 
with e, = 9.6, the dimensions of a 3-dB Lange coupler are W/h — 0.107 and S/h — 0.071, where W 
is the width of each strip and 5 is the spacing between adjacent strips. 

Figure 4.42(b) shows the unfolded Lange coupler with four strips of equal length; it offers the 
same electrical performance but is easier for circuit modeling [79]. The even-mode characteristic 
impedance Z e4 and odd-mode characteristic impedance Z^ of the Lange coupler with Zjj = Z e4 Z o4 
in terms of the characteristic impedances of a two-conductor line (which is identical to any pair of 
adjacent lines in the coupler) can be obtained by 



Zoo + Z, 



lie 



Z„4 = 



3Zo 0 + Zoe 

Zoe + Zoo , 
3Zn e + Zoo 



(4.68) 
(4.69) 



where Z 0e and Z 0o are the even- and odd-mode characteristic impedances of the two-conductor 
pair [80]. 

The midband voltage coupling coefficient C is given by 



196 



POWER TRANSFORMERS, COMBINERS, AND COUPLERS 



The even- and odd-mode characteristic impedances Z 0e and Z 0o , as functions of the characteristic 
impedance Z 0 and coupling coefficient C, are determined by 



Zoe 



Zoo 

For alumina substrate with e r = 9.6, the dimensions of such a 3-dB unfolded Lange coupler are 
W/h — 0.112 and Slh — 0.08, where W is the width of each strip and S is the spacing between the 
strips. 

The design theory for TEM transmission-line couplers is based on an assumption of the same 
phase velocities of the even and odd propagation mode. However, this is not the case for coupled 
microstrip lines, since they have unequal even- and odd-mode phase velocities. In this case, the odd 
mode has more fringing electric field in the air region rather than the even mode with electrical 
field concentrating mostly in the substrate underneath the microstrip lines. As a result, the effective 
dielectric permittivity in the latter case is higher, thus indicating a smaller phase velocity for the 
even mode. Consequently, it is required to apply the phase velocity compensation techniques to 
improve the coupler directivity that decreases with increasing frequency. Figure 4.43(a) shows a 
typical wiggly-line coupler (with sawtooth shape of coupled lines), where wiggling the adjacent 
edges of the microstrip lines, which makes their physical lengths longer, slows the odd-mode wave 
without much affecting the even-mode wave [81]. High directivity can also be achieved by using 
capacitive compensation. Figure 4.43(b) shows the capacitively compensated microstrip directional 
coupler where the two identical lumped capacitors are connected between coupled lines at their edges. 
Physically, these edge capacitors affect the odd mode by equivalent extension of the transmission- 
line electrical lengths, with almost no effect for even mode. For an ideal lossless operation condition 
at 12 GHz using standard alumina substrate, the compensated coupled-line microstrip directional 
coupler can improve directivity from 13.25 dB to infinity [82]. Capacitive compensation can be 
performed by a gap coupling of the open-circuit stub formed in a subcoupled line [83]. In this case, 
the coupler directivity can be improved by 23 dB in a frequency range from 1 to 2.5 GHz compared 
to the directivity of the conventional uncompensated microstrip coupler. 

At radio frequencies and low microwaves, the conventional quarter-wavelength directional cou- 
pler has very large dimensions that limit their practical application especially in monolithic circuits. 
Figure 4.44 shows a reduced-size directional coupler consisting of the two coupled microstrip lines, 




FIGURE 4.43 Coupled-line directional couplers. 



Z 0 



1 + C 4C - 3 + V9 - 8C 2 



1 - C 



2C 

~C4C + 3 - 



8C 2 



1 + C 



2C 



(4.71) 
(4.72) 



REFERENCES 197 



Through 



/< V4 



Coupled 



Input 



FIGURE 4.44 Coupled-line reduced-size directional coupler. 



the electrical lengths of which are much smaller than quarter wavelength. The main problem of the 
coupler at frequencies, where the electrical length of its coupled lines is smaller than quarter wave- 
length, is that the degree of coupling linearly varies with frequency. To compensate this frequency 
behavior, the output port 3 can be connected to a series inductor L followed by a shunt resistor R 
[38,84]. The inductance value depends on the coupling value and flatness, and midband frequency, 
while the resistance value depends on the impedance of the secondary line and inductance value. 
Such a microstrip reduced-size directional coupler with L — 1 80 nH and R — 62 Q can provide the 
coupling of about 30 dB with flatness of ±0.1 dB, directivity greater than 20 dB, insertion lossless 
than 0.25 dB, and VSWR less than 1.15 in a frequency bandwidth of 60% around 200 MHz. Tuning of 
the center bandwidth frequency and coupling can be simply realized by varying the inductance value. 



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Coupler with the High Directivity," IEEE MTT-S Int. Microwave Symp. Dig., vol. 3, pp. 1895-1898, June 
2004. 

84. L. Maloratsky, "Miniature Directional Coupler," U.S. Patent 5424694, June 1995. 



5 Filters 



Historically, an introduction of bandpass electrical filters invented independently by Karl W. Wagner 
and George A. Campbell played a vital role in their successful development in high-frequency wireless 
telegraphy, wireless telephony, or multiplex wire telephony transmitting systems when a maximum 
number of channels could simultaneously operate on a given circuit with a minimum frequency 
interval between adjacent channels [1,2]. In addition to separating the various channels from each 
other in multiplex line, it was found convenient from an operating standpoint to separate, within the 
toll offices, the carrier frequencies as a group (using a high-pass filter) from the frequencies intended 
for ordinary telephony and telegraphy (using a low-pass filter), and transmit them by separate lines [3] . 

This chapter discusses the basic types of radio frequency (RF) and microwave filters based on the 
low-pass or high-pass sections, and bandpass or bandstop transformation. Classical filter design ap- 
proaches, using image parameter and insertion loss methods, are given for low-pass and high-pass LC 
filter implementations. The quarterwave-line and coupled-line section, which are the basic elements 
of microwave transmission-line filters, are described and analyzed. Different examples of coupled- 
line filters including interdigital, combline, and hairpin bandpass filters are given. Special attention 
is paid to microstrip filters with unequal phase velocities that can provide unexpected properties due 
to different implementation technologies. Finally, the typical structures, implementation technology, 
operational principles, and band performance of the filters, based on surface and bulk acoustic waves, 
are presented. 

5.1 TYPES OF FILTERS 

The most general types of electrical filters in terms of the circuit configuration of the basic elements 
can be represented by a ladder structure in the form of a T-network shown in Figure 5.1(a) and a jr- 
network shown in Figure 5.1(b), or a lattice structure shown in Figure 5.1(c), where the corresponding 
reactive elements are represented schematically by conventional blocks [4], A lattice network can be 
realized with any desired amplitude characteristic, and its characteristic impedance is independent of 
its transmission properties. According to the classification in terms of the character of their elements, 
they can be considered as the LC filters, transmission-line filters, or resonator-type filters. 

The low-pass filter (LPF) with the frequency response shown in Figure 5.2(a) passes the wave 
energy in a passband region from zero frequency up to a determined cutoff frequency ideally without 
attenuation and rejects all energy beyond that limit in a stopband region, with a finite transition between 
passband and stopband regions. The high-pass filter (HPF) with the frequency response shown in 
Figure 5.2(b) prevents the transmission of frequencies in a stopband region below a determined point 
(cutoff frequency) and appears to be electrically transparent to frequencies beyond this point ideally 
without attenuation, with a finite transition between stopband and passband regions. 

The bandpass filter (BPF) with the frequency response shown in Figure 5.2(c) passes the wave 
energy in a passband region from certain lower to upper frequency limits ideally without attenuation 
and stops all energy outside these two limits in the stopband regions. The band-reject filter (BRF) 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



201 



202 FILTERS 



(a) 



O 1- 



O 4- 




(c) 

FIGURE 5.1 Filter configurations. 



(h) 



* O 



dB, 



dB 



Transition Slop band 

\ 



Passband 



Stopbatid Transition 

/ 



Cutoff' 



(a) 



l|. Stopbatid 



dB 



Stopbatid 



^ Pass band ^ 



Passband 



Cutoff 



(ft) 



Stopband 



Passband 



Passband 



W id) 

FIGURE 5.2 Filter frequency responses. 



TYPES OF FILTERS 203 




4i y-c 

FIGURE 5.3 Voltage-driven two-port network terminated in its image impedance. 



with the frequency response shown in Figure 5.2(d) is used when a certain unwanted frequency or 
band of frequencies has to be rejected. Outside of the rejection band or stopband, all frequencies will 
pass ideally without attenuation. All-pass filters pass all frequency components of the input signal, but 
introduce a predictable phase shift for its different components. A short impulse on the input side of 
such a filter is modified into a longer frequency-modulated signal at the output. As an example, an all- 
pass filter can be represented by a matched transmission line or LC lattice structure of a different order. 

For an arbitrary voltage-driven two-port network specified by its transmission ABCD-parameters, 
as shown in Figure 5.3, the image impedance Zji is defined as the input impedance at input port 
1 when output port 2 is terminated with the image impedance Z; 2 , which is defined as the input 
impedance at output port 2 when port 1 is terminated with Zu . Thus, both ports are matched when 
terminated in their image impedances that can be expressed through the ABCD-parameters as 




(5.1) 
(5.2) 



or Z i2 = DZu/A, where the factor yJD/A can be interpreted as a transformer turns ratio [5]. If the 
network is symmetric, then A — D and Z i2 = Z\\ . 

The propagation factor for the two-port network can be defined as 



I AD - 



IBC 



(5.3) 



where y — a + jfi is the propagation constant, a is the attenuation constant, and is the phase 
constant. Eq. (5.3) can be rewritten as 



cosh y — -J AD. 



(5.4) 



Table 5.1 lists the image impedances and propagation factors (along with other important param- 
eters) for the two-port T- and it -networks. 

The basic network unit for realizing all-pass prototype filters is a lattice structure, as shown in 
Figure 5.4, where there is a conventional abbreviated representation on the right-hand side. This 
lattice structure is not only symmetric with respect to the two ports, but also balanced with respect 
to ground. If the impedance of the series arm of the lattice is equal to Z\ and the impedance of the 
parallel arm is equal to Z 2 , the image impedance of the lattice structure is defined as 



Zj — si ZiZ 2 



(5.5) 



204 FILTERS 

TABLE 5.1 Image Parameters for T- and jt -Networks. 



7-Network 



7r -Network 



ABCD-parameters 



1 + 



B = Z X + 



2Z 2 
7 2 
4Z 2 



C : 



I 

zi 



D = 1 H - 

2Z 2 

Z-parameters 
Zn 

Zl2 = Z 2 1 = Z2 



Zi 

Z22 = Z2 + — 



Image impedance 

z iT = yzTZ2,/] 

Propagation factor 



4Z 2 



2Z 2 V z 2 



(£) 



ABCD-parameters 
A ■■ 

B - 
C ■■ 



1 + 

Zi 
1 



2Z 2 



Zj 



D = 1 + 



2Z 2 



F-parameters 



1 1 

I'll = ^22 = — + 



Yu = F 2 i 
Image impedance 



Zi 
1 

zT 



N/zTzi 



1 + ■ 



2Z 2 



Z1Z2 

Z iT 



4Z 2 



Propagation factor 



2Z 2 V Z 2 \2Z 2 / 




FILTER DESIGN USING IMAGE PARAMETER METHOD 



205 



and the propagation constant y is derived from 



tanh — = 




(5.6) 



The amplitude response of the lattice filter depends on only the ratio of the branch reactances and 
is independent of the input and output impedances. When Z\ — Z2, then tanh (y/2) = ±1, and a 
peak of attenuation occurs. The position of this peak can be modified without impedance-change by 
multiplying Z[ by a real positive factor and dividing Z 2 by the same factor. Lattice networks must 
be designed and build with great care in order to balance their impedances. Once reactance values 
are chosen to resonate at strategic positions in the passband (such as the cutoffs and the center of 
passband), the level of the reactances between these strategic positions must be maintained in both 
arms in an exactly prescribed fashion. Moreover, the peak of attenuation outside the passband is 
controlled by the level of the impedances of both arms when they are of the same nature (capacitive 
or inductive), whereas the position of the attenuation peak depends on the sharpness of the reactance 
curve at cutoff, which is effectively within the passband. 



5.2 FILTER DESIGN USING IMAGE PARAMETER METHOD 
5.2.1 Constant-^ Filter Sections 

The image attenuation of the filter is completely defined by its poles with their specified orders. If 
the image impedance is of the first order, called the constant-k, the filter design leads to the structure 
where the filter consists of as many sections as there are poles, and each section is designed separately. 
First consider the low-pass T-network shown in Figure 5.5(a) where the series inductors and shunt 
capacitors tend to block high-frequency signals while passing low-frequency signals. For Z\ = jcoL 
and Zi — ilijcoC), from Table 5.1 it follows that the image impedance Z;t can be written as 




(5.7) 



If we introduce and define a cutoff frequency co c as 



2 




(5.8) 



and a characteristic impedance Z 0 as 




(5.9) 



where k is a constant, then Eq. (5.7) can be rewritten as 




(5.10) 



from which it follows that Zrr = Zq for co — 0. 



206 FILTERS 




In this case, the propagation factor given in Table 5.1 can be written as 




(5.11) 



Hence, each filter section is characterized by its Zobel parameter m related to the attenuation pole 
Q — cola> c , defined as 




(5.12) 



For the low-pass 7r-network shown in Figure 5.5(b) when Z { — jcoL and Z 2 = l/(j(oC), the cutoff 
frequency co c , characteristic impedance Zq, and propagation factor exp (y) are the same as for a 
low-pass T-network given by Eqs. (5.8), (5.9), and (5.1 1), respectively. At co — 0, we have that Zjt = 
Z- m — Z Q , where Z- m is the image impedance of the low-pass n -network, but Z iT and Z- m are generally 
not equal at other frequencies. 

Typical phase and attenuation constants for a low-pass constant-fc filter section are shown in 
Figure 5.5(c). In this case, the attenuation constant a is zero in the passband when co < co c , and tends 
to infinity in the stopband asa^ oo. The attenuation rate for co 3> co c is 40 dB/decade. According to 
Eq. (5.1 1), the phase constant fi increases from 0 to it in the passband where the image impedance is 



FILTER DESIGN USING IMAGE PARAMETER METHOD 



207 



2C 



2C 



C 



2L 



[21. 



{a) (b) 
FIGURE 5.6 High-pass constant-i: filter sections. 



real, and becomes constant and equal to it in the stopband where the image impedance is imaginary. 
This type of filter is known as a constant-fc low-pass prototype, with only two parameters to choose 
(L and C), which are determined by the cutoff frequency w c and characteristic impedance Z 0 . 

For high-pass constant-fc sections, the positions of the inductors and capacitors are reversed from 
those in the low-pass prototype. As a result, the characteristic impedance Z 0 of the high-pass constant- 
k filter sections in the form of T- and ^-sections, shown in Figure 5.6, is defined similarly to the 
low-pass constant-^ filter sections given by Eq. (5.9), and its cutoff frequency a> c is calculated from 



1 



2VLC 



(5.13) 



5.2.2 /n-Derived Filter Sections 

The main disadvantage of the constant-^: filter is that the image impedance, which should terminate 
the filter section at both ports, is a function of frequency that is not likely to match a given source 
or load impedance. In addition, its attenuation is sufficiently small near cutoff frequency and slowly 
increases. The problems can be overcome with the modified m-derived filter sections. 

In this case, as shown in Figure 5.7(a), the impedances Z\ and Z 2 in a constant-^ T-section are 
replaced with the modified impedances Z lm and Z 2m as 



Zim = mZi 
1 - r, 

Zlm 



- 



-Zi + 



(5.14) 
(5.15) 



the values of which are chosen to obtain the same value of the image impedance Z iT as for the constant- 
k section given in Table 5.1 [6]. Because the impedances Z[ and Z 2 represent reactive elements, the 



jj/7,/2 



Z-Jm 



1 - m' 



mZ v '2 



\7.Jm 



I - m" 
2m 



FIGURE 5.7 m-derived filter sections. 



I nr_ 

2m 



208 FILTERS 



modified impedance Z 2m represents two elements in series. Note that m = 1 reduces the m-derived 
filter section to the original constant-A:- filter section. 

For a low-pass T-section when Zi = ja>L and Z2 = 1/(/&jC), the modified impedances Zi m and 
Z 2m , given by Eqs. (5.14) and (5.15), can be rewritten as 



Zim = jma>L 
1 - m 2 



Q)L + 



1 

jmcoC 



(5.16) 
(5.17) 



which results in the filter circuit of Figure 5.8(a). In this case, the propagation factor for the ra-derived 
section can be written using Table 5.1 as 



e Y — \ -\ — — ^ — h ./— — ( 1 H — ~~~ 
2Z 2m V Z2m V 4Z 2m 



For the low-pass m-derived T-section, 



Zlm 
Z 2 m 



(D 

2m — 



(5.18) 



(5.19) 



mL/2 



mUl 



2C!m 



2C/m 



■ mC 



1 - m~ 

Am 



Urn 



Am 
1 nr 



ra-derived 
attenuation 



Composite 
response 




0 0>c f:) * 0) 

FIGURE 5.8 Low-pass and high-pass m-derived T-sections and frequency response. 



FILTER DESIGN USING IMAGE PARAMETER METHOD 



209 



where co c = 2/V LC as for the low-pass constant-/: filter section, and 



1 + 



4Z, 



(5.20) 



Substituting Eqs. (5.19) and (5.20) intoEq. (5.18) shows that when 0 < m < 1 , then the propagation 
factor e y is real and \e y \ > 1 for co > co c . Hence, the filter stopband begins at co — co c , similarly to the 
constant-/: section. However, when co — co^ and the filter Zobel parameter m is defined as 



m = J 1 



(5.21) 



the denominators in both Eqs. (5.19) and (5.20) vanish and e y becomes infinite, thus implying infinite 
attenuation. Physically, this pole in the attenuation characteristic is caused by the resonance of the 
series LC resonator at co^ in the shunt arm of the T-section. The infinite attenuation occurs after the 
cutoff frequency co c , and the position of pole can be controlled with the value of m. The best result 
with minimum variation of the image impedance Z iT over the filter passband is achieved when m — 
0.6. The high-pass m-derived T-section is shown in Figure 5.8(£>), for which a> c = i/(2\f~LC) as for 
the high-pass constant-/: filter section and 



m = ,/ 1 



(5.22) 



The low-pass m-derived section shown in Figure 5.7(b) can be obtained from the corresponding 
two m-derived T-sections connected in cascade. In this case, the impedances Z lm and Z 2m in the 
series and shunt arms must be multiplied by a factor of 2. Figure 5.9 shows the (a) low-pass and 
(b) high-pass m-derived n -sections, where m for sharp cutoff is defined by Eqs. (5.21) and (5.22), 
respectively, and equals to 0.6 for optimum matching. The inductor L and capacitor C are the same 
as for the corresponding low-pass and high-pass constant-/: 7r-sections. 

From Figure 5.8(c) it follows that the filter attenuation of the ni-derived section decreases for co > 
(Oaa- Since it is often desirable to have infinite attenuation as co — >• oo, the m-derived section can 
be cascaded with a constant-/: section to give the composite attenuation response. By combining in 
cascade the constant-/: T-section, the m-derived sharp-cutoff T-section, and the m-derived bisected- 
ji matching sections at the filter input and output, the four-stage composite filter with desired 



ml. 



mC/2 



1 - nr 
2m 



d= mC!2 



l-nr 



2m 



2I.ini 



1 - m 



-t 



2!,im 



1 nf 



{«) ((>) 
FIGURE 5.9 Low-pass and high-pass m-derived ^-sections. 



210 FILTERS 



attenuation and matching properties can be realized [5]. The sharp-cutoff section with m < 0.6 places 
an attenuation pole near the cutoff frequency to provide a sharp attenuation response, whereas the 
constant-fc section provides high attenuation further into the stopband. The bisected-7r sections at the 
ends of the filter match the nominal source and load impedance Z 0 to the internal image impedances 
Zix of the constant-A: and m-derived sections. 



5.3 FILTER DESIGN USING INSERTION LOSS METHOD 

Unlike the image parameter method, the insertion loss method allows a high degree of the filter 
performance control over the passband and stopband when the desired amplitude and phase charac- 
teristic can be synthesized with a systematic way. For example, if a minimum insertion loss is most 
important, a binomial filter response could be used. However, a requirement for the sharpest cutoff 
can be satisfied with a Chebyshev filter response, or better phase response can be obtained by using 
a linear phase filter design. In all cases, the insertion loss method allows filter performance to be 
improved in a straightforward manner, at the expense of a higher order filter. 

In the insertion loss method, a filter response is defined by its attenuation A (or insertion loss in 
decibels equal to 101og 10 A), which is defined as the ratio of power available from the source Ps to 
power delivered to the load Pl, 



ft = j_ 

Pl l-\r(co)\ 2 



A = = : — 2 (5.23) 



where T(co) is the reflection coefficient. For the input impedance Z in (a>) = R^ca) + jX in (co), the 
reflection coefficient T(co) can be written as 

„. , Z to (<w)-Z 0 Rm (co) - Z 0 + jXin («) ,_„., 

T (co) — = (5.24) 

Zi„(w) + Z 0 Rm(a>) + Z 0 + jX m (co) 

where Z 0 is the characteristic impedance. From Eq. (5.24) it follows that the magnitude of the 
reflection coefficient is even function of co because 

\r (ut = \r (-io)\ 2 (5.25) 

and it can be expressed as a polynomial of co 2 . As a result, Eq. (5.23) can be rewritten as 



M(co 2 ) 
N (ft> 2 ) 



5.3.1 Maximally Flat Low-Pass Filter 

The low-pass binomial or Butterworth response of the filter (named in honor of S. Butterworth who 
described this response in 1930), which is completely defined by its poles, is specified as 

— J (5.27) 

where /; is the order of the filter, which corresponds to the number of required reactive elements, and 
co c is the cutoff frequency. The insertion-loss function of this filter in the passband has the flattest 
possible shape in the passband (maximally flat), having the maximum number of (2n — 1) zero 
derivatives at co — 0, and is a monotonically increasing function. The passband extends from co — 0 



FILTER DESIGN USING INSERTION LOSS METHOD 



211 



Slopband 




(b) 

FIGURE 5.10 Maximally flat responses and low-pass filter prototype. 



to a) — co c , with the power loss ratio of (1 + k 2 ) at the band edge and k — 1 for a commonly used 
maximum passband attenuation of 3 dB. Above the cutoff frequency, the attenuation increases almost 
linearly that gives a rate of 6n dB/octave on a logarithmic frequency scale. The attenuation functions 
of the low-pass Butterworth filter are sketched in Figure 5.10(a) for small (n — 2) and large (n = 
5) filter orders. It is clearly seen that the larger order provides a better approximation to the ideal 
low-pass response. 

As a design example, consider the two-element low-pass filter prototype shown in Figure 5.10(6) 
and derive the normalized element values of an inductor L\ and a capacitor C 2 for a maximally flat 
response, assuming a source impedance Rs — Zq — 1 Q and a cutoff frequency <w c = 1 radian/s. For 
n — 2, the input impedance of this filter can be written as 



1 - jcoCiR L 
1 + {coC 2 Rif 



(5.28) 



Then, substituting Eq. (5.28) to Eq. (5.23), and using Eq. (5.24), results in 



A= 1 + 



(1 - R h ) 2 + (R[Cj + L\- 2LiC 2 Rl) co 2 + {LiC 2 R L ) 2 



4Rr 



(5.29) 



212 FILTERS 



R< J = gO=\ L 2 = 82 

■AM O • f~Y"Y"Y"\ 




£3= S3 



Go Kb 1 



< -2 = 82 



givl 



(7.) 



FIGURE 5.11 Ladder circuits for low-pass filter prototypes. 



which represents a polynomial of co 2 . Comparing to Eq. (5.27) for k — 1 and n — 2 shows that — 
1 Q, since A — 1 for a — 0. In this case, the coefficient of or must be equal to zero, which results in 
L\ — Ci - Then, for the coefficient of a> 4 to be unity, L\ — C2 — V2. 

In principle, this design procedure can be extended to find the element values for filters with an 
arbitrary numbers of elements, but analytically it becomes too complicated. Generally, by constructing 
the rational transfer function from Eq. (5.27), the values of elements (normalized capacitors and 
inductors) shown in Figure 5.11 are calculated from 

go = 1.0 

fli - 1 \ 

gi = 2sin I ji I fori = 1, . . , , n (5.30) 

V 2;i J 

gn+i = 1.0 

which are given in numerical form in Table 5.2 [7,8]. For convenience, the element values for 
maximally flat low-pass filter prototypes are given for n — 1 to 8, and these data can be used with 



TABLE 5.2 Element Values for Maximally Flat Low-Pass Filter Prototypes. 



// 


gl 


g2 


S3 


g4 


g5 


gs 


gi 


?8 


g9 


1 


2.0000 


1.0000 
















2 


1.4142 


1.4142 


1.0000 














3 


1.0000 


2.0000 


1.0000 


1.0000 












4 


0.7654 


1.8478 


1.8478 


0.7654 


1.0000 










5 


0.6180 


0.6180 


2.0000 


0.6180 


0.6180 


1.0000 








6 


0.5176 


1.4142 


1.9318 


1.9318 


1.4142 


0.5176 


1.0000 






7 


0.4450 


1.2470 


1.8019 


2.0000 


1.8019 


1.2470 


0.4450 


1.0000 




8 


0.3902 


1.1111 


1.6629 


1.9615 


1.9615 


1.6629 


1.1111 


0.3902 


1.0000 



FILTER DESIGN USING INSERTION LOSS METHOD 



213 



either of the ladder circuits of Figure 5.11, where R s = R Q and Gs = Gq. The two-port Butterworth 
filters considered here represent the symmetrical network structure when g 0 — g n +i, gi — gn, and so 
on. If a termination resistance higher than 1 n is desired, the reactances are multiplied accordingly. 
The reactances are then scaled so that they have the same value at the new desired cutoff frequency a> c . 

The elements alternate between series and shunt connections, and go and g n+ i represent the source 
and load resistances for a network in Figure 5.11(a) and the source and load conductances for a 
network in Figure 5.11 the reactive components are calculated from 

gi ?i 

Q = Lj = ^-^ (5.31) 



and both low-pass ladder circuits of Figure 5.1 1 give the same frequency response. 

The degree of a Butterworth low-pass filter prototype defined by a minimum stopband attenuation 
at co — &> s and a maximum passband attenuation A max at a> — oj c can be derived from Eq. (5.27) 
in the form 



. A min 1 

log, 0 J- — j 

n > Ag^_ ( g 32) 

2 logm- 
en,. 



For example, if a minimum attenuation in the stopband A m i n = 10 4 (or 40 dB) at chJcd c — 2 for 
A max — 2 (or 3 dB) is required, then n > 6.644, and a 7-pole (n — 7) Butterworth prototype should be 
chosen. 

In the high-pass ladder filters, which are dual of the low-pass ladder filters, the series inductors are 
replaced with series capacitors, while shunt capacitors are replaced by shunt inductors. The reactances 
are the same at the normalized cutoff frequency oj c — 1 radian/s and Rq — 1 f2. The Butterworth 
high-pass filter prototype is described by the following denormalization of the normalized low-pass 
data from Table 5.2: 

1 R 0 

Cj = L, = — . (5.33) 

gi^c^O gi«c 

The Butterworth approximation is useful for many applications; however, its main advantage is 
its mathematical simplicity. The low-pass Butterworth response was derived on the assumption that 
behavior at zero frequency was far more important than behavior at any other frequency. This leads to 
a class of filters with good phase response and tolerably good amplitude response but with very poor 
characteristics around the cutoff frequency. The Butterworth function is unsuitable for applications 
that require uniform transmission of frequencies in the passband and sharp rise at cutoff. 



5.3.2 Equal-Ripple Low-Pass Filter 

By changing the approximation conditions, it is possible to obtain much better characteristics near 
cutoff frequency. Besides, it may be required that all frequencies in the passband are equally important, 
and that it is desirable to minimize the maximum deviation from the ideal response. The Chebyshev 
approximation can exhibit the equal-ripple passband and maximally flat stopband when the attenuation 
of the low-pass filter is written as 



A = i+ * 2r "(£) 



(5.34) 



214 FILTERS 



where k is the ripple constant and T n (co/co c ) is the a Chebyshev function of the first kind of order n, 
which is defined as 



_. CO \ CO 

cos n cos — I for — < 1 



COr 

(5.35) 

CO 

cosh I n cosh — | for — > 1 

CO c 



( — I <*>\ 

I n cosh — 



From Eq. (5.35) is not clear that T n (co/co c ) is actually the nth-order polynomial. However, by 
applying simple trigonometric identities, the first four Chebyshev polynomials are written as 

Ti(x) = x (5.36) 

T 2 (x) = 2x 2 - 1 (5.37) 

T 3 (x) = 4x 3 - 3x (5.38) 

T 4 (x) = Sx 4 - Sx 2 + 1 (5.39) 

with higher order polynomials found using general recurrence relation, 

T D (x) = 2xT n _, (x) - r n _ 2 (x) (5.40) 

where x — co/co c . 

Since the Chebyshev polynomials have the property that T n (0) = 0 for odd n and T n (0) = 1 for 
even n, from Eq. (5.34) it follows that the equally-ripple low-pass filter have a unity attenuation at 
(o = 0 for odd n, but an attenuation of (1 + k 2 ) at co — 0 for even n. In the passband, the attenuation 
response varies between the values of zero and A max = 1 + k 2 , as seen from Figure 5.12(a) for 
n — 2, 3. Above the cutoff frequency co c , the attenuation curve rises monotonically. The rate of 
increase depends not only upon the number of poles or resonators but also upon such a design 
parameter as the height of the ripples: the attenuation rate is higher for larger passband ripples. 

For the two-element low-pass filter prototype, shown in Figure 5.12(b) with a source impedance 
R s — R 0 — 1 f2 and a cutoff frequency co c — 1 radian/s, equating Eq. (5.34) to Eq. (5.29) and using 
Eq. (5.37) result in 

1 | fcW ^ I 1) 1 I U-^ L ) 2 + (^C| + L ? -2L 1 C 2 ^)^ + (L 1 C 2 ^^ 

(5.41) 

which can be solved to define ^l, Li, and C2 if the ripple determined by k 2 is known. Since k 2 — 
(1 — R L ) 2 /(4R L ) at co — 0, then equating coefficients of co 2 and co 4 yields the additional equations, 

t ( L 1 Ct R\ ) 2 

4k 2 = (5.42) 
4i? L 

-4k 2 = 1 (5.43) 

4R L 

which are necessary to find L\ and C 2 . Note that a value for is not unity for even n, thus resulting 
in an impedance mismatch if the load represents a unity normalized impedance. 

Because it is too complicated to find analytically element values for filters with arbitrary numbers 
of elements, by constructing the rational transfer function from Eq. (5.34), the values of elements 



FILTER DESIGN USING INSERTION LOSS METHOD 



215 



1-A 2 





Passband 


l 


Stopband 






— »r* 

i 

n-3 . 








I 

1 // 
1 if 
1/ 

r 


n-2 






J 


+■ 



tO/tOc 




FIGURE 5.12 Equal-ripple responses and low-pass filter prototype. 



(normalized capacitors and inductors) shown in Figure 5.1 1 can be calculated from 



go = 1.0 
2 

gi = -sin., 
y \2n 



(s) 



/2/-1 \ /2i — 3 

4 sin 7r sin n 

1 V 2« / V 2/i 



r- 



for i = 2, 3, . . . , n 



(5.44) 



with 



^ n+1 = 1.0 for odd n and g n+1 = coth — for even n 



y = sinh — /S = In 
2n 



coth 



A(dB) 
17.3718 



where A(dB) is the passband ripple in decibels [7]. Some typical element values for such filters are 
given in numerical form in Table 5.3 for various passband ripples A(dB) and for the filter degree of 
n = 1 to 7. 



216 FILTERS 

TABLE 5.3 Element Values for Equal-Ripple Low-Pass Filter Prototypes. 



Ripple 



(OB) 


II 


Si 


g2 


£3 


g4 


g5 


g6 


gl 


gs 


0.1 


1 


0.3052 


1.0000 
















2 


0.8431 


0.6220 


1.3554 














3 


1.0316 


1.1474 


1.0316 


1.0000 












4 


1.1088 


1.3062 


1.7704 


0.8181 


1.3554 










5 


1.1468 


1.3712 


1.9750 


1.3712 


1.1468 


1.0000 








6 


1,1681 


1.4040 


2.0562 


1.5171 


1.9029 


0.8618 


1.3554 






7 


1.1812 


1.4228 


2.0967 


1.5734 


2.0967 


1.4228 


1.1812 


1.0000 


0.5 


1 


0.6986 


1.0000 
















2 


1.4029 


0.7071 


1.9841 














3 


1.5963 


1.0967 


1.5963 


1.0000 












4 


1.6703 


1.1926 


2.3661 


0.8419 


1.9841 










5 


1.7058 


1.2296 


2.5408 


1.2296 


1.7058 


1.0000 








6 


1.7254 


1.2479 


2.6064 


1.3137 


2.4758 


0.8696 


1.9841 






7 


1.7372 


1.2583 


2.6381 


1.3444 


2.6381 


1.2583 


1.7372 


1.0000 


3.0 


1 


1.9953 


1.0000 
















2 


3.1013 


0.5339 


5.8095 














3 


3.3487 


0.7117 


3.3487 


1.0000 












4 


3.4389 


0.7483 


4.3471 


0.5920 


5.8095 










5 


3.4817 


0.7618 


4.5381 


0.7618 


3.4817 


1.0000 








6 


3.5045 


0.7685 


4.6061 


0.7929 


4.4641 


0.6033 


5.8095 






7 


3.5182 


0.7723 


4.6386 


0.8039 


4.6386 


0.7723 


3.5182 


1.0000 



The degree of a Chebyshev low-pass filter prototype, defined by a minimum stopband attenuation 
Amin at a> — a> s and a maximum passband ripple A max at co — a; c , can be derived from Eqs. (5.34) and 
(5.35) in the form 



cosh 
n > 




— _1 w s 

cosh — 

For instance, using the same example as given for the Butterworth low-pass filter prototype with 
a minimum attenuation in the stopband A m \ D — 10 4 (or 40 dB) at a>Ja) c — 2, but for a passband ripple 
Amax — 1-0233 (or 0.1 dB) for the Chebyshev response, results in n > 5.45. Hence, a 6-pole (n — 6) 
Chebyshev prototype should be chosen to meet this specification that demonstrates the superiority of 
the Chebyshev design over the Butterworth design. 

The Chebyshev function is extremely useful in applications where the magnitude of the transfer 
function is of primary concern. This approximation gives more constant magnitude response through- 
out the passband and faster rate of cutoff outside the passband. As a consequence, the transition range 
for reaching a prescribed is a minimum, and the attenuation in the stopband is never less than 
that prescribed attenuation. However, the phase response of the Chebyshev filter tends to be poor, 
with a rapid increase in the group-delay variations at the band edges. 



5.3.3 Elliptic Function Low-Pass Filter 

The maximally flat and equal-ripple responses both have monotonically increasing attenuation in 
stopband. However, in many applications it is required to specify a minimum stopband attenuation, 
in which case a better cutoff rate can be obtained. By changing the approximation conditions, it is 
possible to obtain much better characteristics near cutoff frequency. This can be achieved by using 
elliptic function filters having equal-ripple responses both in the passband and the stopband, as shown 



FILTER DESIGN USING INSERTION LOSS METHOD 



217 




FIGURE 5.13 Elliptic function low-pass filter response. 



in Figure 5.13, where A max is the maximum attenuation in the passband and A m ; n is the minimum 
attenuation in the stopband. Elliptic function filters are also known as Cauer or Zolotarev filters, 
which achieve the smallest filter order for the same specifications, or the narrowest transition width 
for the same filter order, as compared to other filter types [4,9]. 

The attenuation of the elliptic function low-pass filter is written as 



(5.46) 



where k is the ripple constant and T n (wlco c ) is chosen so that it has an equal-ripple attenuation in 
the passband and the stopband. Depending on whether it is even or odd, the polynomial function 
T n (co/a) c ) represents one of two following forms: 



n/2 

n 



M - 



CO 
0)c 



n/2 

n 

>=t 

(n-D/2 

n 



for even n 



N - 



(5.47) 



(n-D/2 

n 



for odd n > 3 



where 0 < coi/a> c < 1 and &> s /cu c > 1 represent some critical frequencies (stopband begins with 
frequency a; s ), M and N are constants to be defined, and T n (a)/a> c ) must lie between the limits — 1 and 
+ 1 in the passband and should take the maximum possible absolute values for the given degree of n 
in the stopband [10,11]. 

Figure 5.14 shows two commonly used network structures for elliptic function low-pass filter 
prototypes. The series-parallel resonant circuits in Figure 5.14(a) are introduced for realizing the 
finite-frequency zeroes to block the signal transmission at these frequencies by means of infinite 
open-circuit conditions. In this case, gi represents the normalized capacitance of a shunt capacitor 
for odd i, while gj is the normalized inductance of an inductor and the primed gj is the normalized 
capacitance of a capacitor in a parallel resonant circuit for even n. For the dual network structure 
shown in Figure 5.14(b), the shunt branches of series resonant circuits are used for implementing 



218 FILTERS 




FIGURE 5.14 Elliptic function low-pass filter prototypes. 



the finite-frequency zeroes to short out the signal transmission at these frequencies by means of the 
short-circuit conditions. In this case, g\ represents the normalized inductance of a series inductor 
for odd i, while gi is the normalized capacitance of a capacitor and the primed g- is the normalized 
inductance of an inductor in a series resonant circuit for even n. Note that either low-pass filter 
structure form can be used because both give the same response. 

Unlike Butterworth and Chebyshev low-pass filter prototypes, there is no simple formula available 
for determining element values of the elliptic function low-pass filter prototypes. Table 5.4 tabulates 



TABLE 5.4 Element Values for Elliptic Function Low-Pass Filter Prototypes. 



11 


co s /co c 




81 


82 




gi 


g4 


84 


gs 


3 


1.4493 


13.5698 


0.7427 


0.7096 


0.5412 


0.7427 










1.6949 


18.8571 


0.8333 


0.8439 


0.3252 


0.8333 










2.0000 


24.0012 


0.8949 


0.9375 


0.2070 


0.8949 










2.5000 


30.5161 


0.9471 


1.0173 


0.1205 


0.9471 








4 


1.2000 


12.0856 


0.3714 


0.5664 


1.0929 


1.1194 


0.9244 








1.2425 


14.1259 


0.4282 


0.6437 


0.8902 


1.1445 


0.9289 








1.2977 


16.5343 


0.4877 


0.7284 


0.7155 


1.1728 


0.9322 








1.3962 


20.3012 


0.5675 


0.8467 


0.5261 


1.2138 


0.9345 








1.5000 


23.7378 


0.6282 


0.9401 


0.4073 


1.2471 


0.9352 








1.7090 


29.5343 


0.7094 


1.0688 


0.2730 


1.2943 


0.9348 








2.0000 


36.0438 


0.7755 


1.1765 


0.1796 


1.3347 


0.9352 






5 


1.0500 


13.8785 


0.7081 


0.7663 


0.7357 


1.1276 


0.2014 


4.3812 


0.0499 




1.1000 


20.0291 


0.8130 


0.9242 


0.4934 


1.2245 


0.3719 


2.1350 


0.2913 




1.1494 


24.5451 


0.8726 


1.0083 


0.3845 


1.3097 


0.4991 


1.4450 


0.4302 




1.2000 


28.3031 


0.9144 


1.0652 


0.3163 


1.3820 


0.6013 


1.0933 


0.5297 




1.2500 


31.4911 


0.9448 


1.1060 


0.2694 


1.4415 


0.6829 


0.8827 


0.6040 




1.2987 


34.2484 


0.9681 


1.1366 


0.2352 


1.4904 


0.7489 


0.7425 


0.6613 




1.4085 


39.5947 


1.0058 


1.1862 


0.1815 


1.5771 


0.8638 


0.5436 


0.7578 




1.6129 


47.5698 


1.0481 


1.2416 


0.1244 


1.6843 


1.0031 


0.3540 


0.8692 




1.8182 


54.0215 


1.0730 


1.2741 


0.0919 


1.7522 


1.0903 


0.2550 


0.9367 




2.000 


58.9117 


1.0876 


1.2932 


0.0732 


1.7939 


1.1433 


0.2004 


0.9772 



FILTER DESIGN USING INSERTION LOSS METHOD 



219 



some useful design data for equally terminated (g 0 = g n+ i = 1) elliptic function low-pass filter 
prototypes representing two equivalent structures shown in Figure 5.14, where normalized element 
values are given for various cojco c and minimum stopband attenuation A m i n for a maximum passband 
ripple A max = 0.1 dB [12]. In this case, a smaller coJco c means a higher selectivity of the filter at the 
expense of reducing stopband rejection. 

Considering the same example as used for the Butterworth and Chebyshev low-pass filter pro- 
totypes with a minimum attenuation in the stopband A min = 40 dB at cojco, = 2 and a maximum 
passband ripple A max = 0.1 dB, from Table 5.4 it follows that a 5-pole (n = 5) elliptic function 
filter prototype should be chosen that demonstrates its superiority over both the Butterworth and 
Chebyshev designs for this type of specification. Note that the even-degree Zolotarev low-pass filter 
has a mismatch at zero frequency and is realizable as an impedance-transforming LC ladder network 
with unequal terminations [13]. 

5.3.4 Maximally Flat Group-Delay Low-Pass Filter 

A maximally flat group-delay low-pass filter prototype is characterized by its transfer (or gain) 
function G(jcolco c ) that is written in the form 

/ co\ an 
G j-) = -„ —T (5-48) 



10, 



k=0 



0->r 



where 



(2n - k) ! 

flk = y-m (n-m (5 - 49) 

represents the coefficients of the reverse Bessel polynomials and &> c is a frequency chosen to give 
the desired cutoff frequency. This transfer function results in better responses in the time domain and 
approaches the ideal Gaussian curve as the degree of approximation is increased (n — »■ oo). It also 
contributes to a group delay that has maximum possible number of zero derivatives with respect to co 
at co — 0, resulting in a maximally flat group delay around co — 0, and is in a sense complementary 
to the Butterworth response, which has a maximally flat amplitude. This maximally flat group-delay 
approximation was originally derived by W. E. Thompson, and for these reasons the filter prototypes 
of this type are also called Bessel-Thompson filters [14]. The first four Bessel polynomials with 
respect to a loss function H(jcolco c ) — \IG{jcolco,) are written as 



H(s) = 



s + 1 for n = 1 

s 2 + 3s + 1 for n = 2 

s 3 + 6s 2 + 15.5 + 15 for n = 3 1 j 

s A + 10s 3 + 45s 2 + 105s + 105 for n = 4 



where s — jcolco c is the normalized complex frequency variable. 

Figure 5.15 shows the typical maximally flat group-delay responses in terms of attenuation 
and group delay for n — 3 and n — 5 that are properly obtained from Eq. (5.48). In general, the 
Bessel-Thompson filters have a poor selectivity, as can be seen from the attenuation curves in 
Figure 5.15(a). However, its selectivity improves according to 



A(dB) = 101og 10 exp 



1 / co 
In — 1 V co. 



(5.51) 



with increasing order n of the filter [15]. From Eq. (5.51), the 3-dB frequency bandwidth can be 
determined as 



CO, 



j(2n - l)ln2 (5.52) 



220 FILTERS 





1 «i/e> c 0 1 

la) (fc) 

FIGURE 5.15 Maximally flat group-delay response. 



(iV <i) t 



approximation of which is sufficiently accurate for n > 3. Hence, unlike the Butterworth response, 
the 3-dB bandwidth of a Bessel-Thompson filter is a function of the filter order when the higher the 
filter order, the wider the 3-dB bandwidth. 

However, the Bessel-Thompson filters have a quite flat group delay in the passband, as shown in 
Figure 5.15(b) for a normalized group delay T g /ro, where To is the delay at zero frequency, and this 
normalized group delay is inversely proportional to the bandwidth of the passband. With increasing 
the filter order n, the group delay is flat over a wider frequency range. Therefore, a high-order 
Bessel-Thompson filter is usually used for achieving a flat group delay over a large passband. 

As an example, consider the gain function G(s) for a third-order Bessel-Thompson low-pass filter 
prototype derived from Eq. (5.48) as 



G(*) = 



15 



s 3 + 6s 2 + 15.s + 15 



(5.53) 



In this case, the attenuation A = \H (s)\ = 1/ | G (s)| can be written by assuming co c = 1 radian/s 



A = 1 + — + — + 

5 75 225 



and the phase 4> — arg[G(s)] as 



i (&>) = — tan 



I5a> — co 3 
15 - 6ar 



(5.54) 



(5.55) 



The group delay, which is a measure of the transit time of a signal through a particular transmitting 
structure versus frequency, is then 



0(£«) 



6a/ + 45g/ + 225 



s m co 6 + 6co 4 + 45m 2 + 225 ' 

Finally, the Taylor series expansion of the group delay is 

CO 6 w g 

T„ = 1 1 1 

8 225 1125 



(5.56) 



(5.57) 



FILTER DESIGN USING INSERTION LOSS METHOD 221 



TABLE 5.5 Element Values for Maximally Flat Group-Delay Low-Pass Filter Prototypes. 



11 


si 




&i 


if | 


5 5 


.fO 


Qn 


So 




1 


2.0000 


1 .0000 
















2 


1.5774 


0.4226 


1.0000 














3 


1.2550 


0.5528 


0.1922 


1.0000 












4 


1.0598 


0.5116 


0.3181 


0.1104 


1.0000 










5 


0.9303 


0.4577 


0.3312 


0.2090 


0.0718 


1.0000 








6 


0.8377 


0.4116 


0.3158 


0.2364 


0.1480 


0.0505 


1.0000 






7 


0.7677 


0.3744 


0.2944 


0.2378 


0.1778 


0.1104 


0.0375 


1.0000 




8 


0.7125 


0.3446 


0.2735 


0.2297 


0.1867 


0.1387 


0.0855 


0.0289 


1.0000 



where the two terms in w 1 and co 4 are zero, resulting in a very flat group delay at co — 0. This is 
the greatest number of terms that can be set to zero, since there are a total of four coefficients in the 
third-order Bessel polynomial, requiring four equations in order to be defined. One equation specifies 
that the gain be unity at co — 0 and a second equation specifies that the gain be zero at co — oo, leaving 
two equations to specify two terms in the series expansion to be zero. This is a general property of the 
group delay for a Bessel-Thompson filter of order n: the first (n — 1) terms in the series expansion of 
the group delay will be zero, thus maximizing the flatness of the group delay at co — 0. 

Design element values for maximally flat group-delay low-pass filter prototypes with g 0 — 1 
derived for the ladder circuits of Figure 5.11 for different values of n = 1 to 8 are given in Table 5.5. 

Figure 5.16 shows the frequency transfer characteristics of the Butterworth, Chebyshev, and 
Bessel-Thompson low-pass filter prototypes based on the ladder circuit shown in Figure 5.11(a) 
and the elliptic low-pass filter prototype based on the circuit structure of Figure 5.14(a). Each filter 
configuration was chosen to have an order of n — 5, with a passband ripple of 0.5 dB for the Chebyshev 




222 FILTERS 



TABLE 5.6 Filter Ranking. 



Filter Type 



Gain Roll-Off 



Linear Phase 



Bessel-Thompson 
Butterworth 
Chebyshev 
Elliptic 



Worst 
Poor 
Better 
Best 



Best 
Better 
Poor 
Worst 



low-pass filter and passband ripple of 0.1 dB for the elliptic low-pass filter. Note that attenuation of 
the elliptic filter can be increased to more than 50 dB starting from wJco c — 1 .7, as it follows from 
Table 5.4. 

Table 5.6 shows a rank ordering of the filters in terms of both gain and phase performance. In 
this case, the elliptic filter offers the very best standard approximation to the ideal low-pass filter 
amplitude behavior, but its group delay deviates considerably from a constant. On the other end, the 
Bessel-Thompson filter provides excellent phase performance, but a quite high order is required to 
achieve a reasonable gain characteristic. If both excellent gain and phase characteristics are abso- 
lutely necessary, two approaches are possible. One either uses computer optimization techniques to 
simultaneously approximate gain and phase, or one uses elliptic filter with excellent gain performance 
followed by the phase compensation circuit with an inverse phase characteristic. 

5.4 BANDPASS AND BANDSTOP TRANSFORMATION 

Low-pass filter prototypes can be transformed to have the bandpass or bandstop responses. This 
transformation into a passband response can be obtained using the frequency substitution in the form 



where co 0 is the center bandwidth frequency, Aco — a> c+ — &> c _ is the passband, and <w c _ and a> c+ are 
the low and high cutoff edges of the passband. In this case, the passband Chebyshev filter is realized 
with the same equal-ripple level, as shown in Figure 5.17. 

As a result, a series inductor L : is transformed into a series resonant circuit with inductor L[ and 
capacitor C[ according to 




(5.58) 




(5.59) 



A 



A 



1 -*f 





0 



0 



an 



FIGURE 5.17 Bandpass frequency transformation for equal-ripple low-pass filter prototype. 



BANDPASS AND BANDSTOP TRANSFORMATION 



223 



where 

L; , Aco 

L[= — C/=^— . (5.60) 

Aco (OqL^ 

Similarly, a shunt capacitor d is transformed into a shunt resonant circuit with inductor L[ and 
capacitor C[ as 

COn ( CO COn \ , 1 

cod = — ( ) Q = wCj' (5.61) 

A&> \con co J coL' 



where 



AtO 1 fOgCi 



t?= 7 i L S=^- ( 5 - 62 ) 



The low-pass filter prototype will be transformed to the bandpass filter when all its series elements 
are replaced by the series resonant circuits and all its parallel elements are replaced by the parallel 
resonant circuits, where each of them are tuned to the center bandwidth frequency coq. In this case, 
the elements in the series resonant circuit of a bandpass filter can be calculated from Eq. (5.60) with 
impedance scaling as 

KiRo , Act) 

V, = ^ C = -= (5.63) 

Aco co^giRo 

whereas the elements in the shunt resonant circuit of a bandpass filter can be calculated from Eq. 
(5.62) with impedance scaling as 



AcoR 0 

AcoR 0 1 co\g\ 



C,'=T^r (5-64) 



where i is an element serial number for low-pass prototype filter and g; is the appropriate coefficient 
given by Table 5.2 for maximally flat low-pass filter prototype or Table 5.3 for equal-ripple low-pass 
filter prototype. 

As an example, let us design a third-order (n — 3) bandpass filter with the center frequency of 2 
GHz and bandwidth of 20%, having a 0.5-dB equal-ripple response and an impedance Rq — 50 Q. 
From Table 5.3, the element values for the low-pass prototype circuit shown in Figure 5.18(a) are 
given by 

gi = 1.5963 g 2 = 1.0967 
g 3 = 1.5963 g 4 = 1.0000. 

Then, the impedance-scaled and frequency-transformed element values for the passband filter 
circuit shown in Figure 5. 18(fo) using Eqs. (5.63) and (5.64) are 

L'j = L' 3 = 3.76 nH C[ = C 3 = 0.1994 pF 
L' 2 = 0.7256 nH C 2 = 8.727 pF. 

Finally, the resulting amplitude response of the Chebyshev passband filter in a frequency domain 
is shown in Figure 5.18(c) where the out-of-band suppression of 40 dB is achieved below 1.3 GHz 
and above 3.0 GHz. 

The inverse transformation that can be applied to obtain a bandstop response is written as 

Aco f CO COn\ 1 
to -+ J (5.65) 

CO 0 \CO 0 CO J 



224 FILTERS 



ii=gi L)=g) 




FIGURE 5.18 Bandpass filter circuit and its amplitude response. 



where m 0 and Ato have the same definition as in Eq. (5.58). In this case, a series inductor L ; of the 
low-pass filter prototype is transformed into a parallel resonant circuit with inductor L[ and capacitor 
C[ according to 

L = C = — — (5.66) 

and a shunt capacitor C\ of the low-pass filter prototype is transformed into a series resonant circuit 
with inductor L[ and capacitor C[ as 

c,'=^ l \=it-f (5 ' 67) 



with further denormalization similar to that given by Eqs. (5.63) and (5.64) for a passband filter. 



TRANSMISSION-LINE LOW-PASS FILTER IMPLEMENTATION 



225 



5.5 TRANSMISSION-LINE LOW-PASS FILTER IMPLEMENTATION 

The design approaches for the filters with lumped elements generally work well at sufficiently low 
frequencies or in small-size monolithic integrated circuits. However, the lumped elements such as in- 
ductors and capacitors are difficult to implement at microwave frequencies where they can be treated as 
distributed elements, and distances between the filter components are not negligible. The other prob- 
lem is that the quality factors for inductors are sufficiently small, thus contributing to additional losses. 



5.5.1 Richards's Transformation 

Generally, the design of a practical distributed filter is based on some approximate equivalence be- 
tween lumped and distributed elements, which can be established by applying a Richards's transforma- 
tion [16]. This implies that the distributed circuits composed of equal-length open- and short-circuited 
transmission lines can be treated as lumped elements under the transformation 

710} 

s = jtan (5.68) 

2co 0 

where s — jco/aj c is the conventional normalized complex frequency variable and co 0 is the radian 
frequency for which the transmission lines are a quarter wavelength [12]. 

As a result, the one-port impedance of a short-circuited transmission line corresponds to the 
reactive impedance of a lumped inductor Z L as 

ICQ) 

Zl = sL — jcoL — _/Ltan . (5.69) 

2m 0 

Similarly, the one-port admittance of an open-circuited transmission line corresponds to the 
reactive admittance of a lumped capacitor Yq as 

Y c = sC = jcoC = jC tan — . (5.70) 
2a>o 

The results given in Eqs. (5.69) and (5.70) show that an inductor can be replaced with a short- 
circuited stub of electrical length 6 — jtcol(2a>o) and characteristic impedance Zo = L, while a capacitor 
can be replaced with an open-circuited stub of electrical length 0 — tccoI(2o)( I ) and characteristic 
impedance Z 0 = 1/C when a unity filter characteristic impedance is assumed. 

From Eq. (5.68) it follows that, for a low-pass filter prototype, the cutoff occurs when co — oj c , 
resulting in 

7r co- 
tan = 1 (5.71) 

2ct> 0 

that gives a stub length 0 — 45° (or 7r/4) with co c — a> 0 /2. Hence, the inductors and capacitors of 
a lumped-element filter can be replaced with short-circuited and open-circuited stubs, as shown in 
Figure 5.19. Since the lengths of all stubs are the same and equal to A/8 at the cutoff frequency co c , 
these lines are called the commensurate lines. At the frequency a> — coo, the transmission lines will 
be a quarter-wavelength long, resulting in an attenuation pole. However, at any frequency away from 
a> c , the impedance of each stub will no longer match the original lumped-element impedances, and 
the filter response will differ from the desired filter prototype response. Note that the response will 
be periodic in frequency, repeating every 4&> c . 

Since the transmission line generally represents a four-port network, it is very convenient to use 
a matrix technique for a filter design. In the case of cascade of several networks, the rule is that the 
overall matrix of the new network is simply the matrix product of the matrices for the individual 



226 FILTERS 



r 



o 



r 4 



o— 



9 = 45° 



9 = 45° 



1-^ 



FIGURE 5.19 Equivalence between lumped elements and transmission lines. 



networks taken in the order of connection [17]. In terms of a Richards's variable, an ABC£>-matrix 
for a transmission line with the characteristic impedance Z 0 can be written as 



A B 
C D 



1 



l 

s 

L Zo 



(5.72) 



representing a unit element which has a half-order transmission zero at s — ± 1 . The matrix of the unit 
element is the same as that of a transmission line of electrical length 8 and characteristic impedance 
Zo. Unit elements are usually introduced to separate the circuit elements in transmission-line filters, 
which are otherwise located at the same physical point. 



5.5.2 Kuroda Identities 

The application of a Richards's transformation provides a sequence of short-circuited and open- 
circuited stubs, which are then converted to a more practical circuit implementation. This can be 
done based on a series of equivalent circuits known as Kuroda identities, which allows these stubs 
to be physically separated, transforming the series stub into the shunt and changing impractical 
characteristic impedances into more realizable impedances [18]. The Kuroda identities use the unit 
elements, and these unit elements are thus commensurate with the stubs used to implement inductors 
and the capacitors of the prototype design. Connecting the unit element with characteristic impedance 
Z 0 to the same load impedance Z 0 does not change the input impedance. The four Kuroda identities are 
illustrated in Figure 5.20, where the combinations of unit elements with the characteristic impedance 
Zo and electrical length 9 — 45°, the reactive elements, and the relationships between them are given. 

To prove the equivalence, consider two circuits of identity at the first row in Figure 5.20 when 
ABCD-matrix for the entire left-hand circuit can be written as 



A B 
C D 



1 



1 o 
sC 1 



1 



1 



l 



s C + 



l + s 2 Z l C 



(5.73) 



where Z\ is the characteristic impedance of the left-hand unit element. 



TRANSMISSION-LINE LOW-PASS FILTER IMPLEMENTATION 



227 




Z„(w- \ )/n 



Z, = Zo/n 



(6) 






11= ] + Z</L 



\/Zd(n- ])n 





n=\ + \/ZoC 



FIGURE 5.20 Four Kuroda identities. 



Similarly, for the right-hand circuit, 



A B 
C D 



1 sZ 2 
s 

— 1 
L z 2 



1 sL 
0 1 



1 s(Z 2 + L) 
s 2 L 



s 

Z~2 



1 + 



where Z 2 is the characteristic impedance of the right-hand unit element. 
The results in Eqs. (5.73) and (5.74) are identical if 

1 1 L 

Zi = Z 2 + L — + C=— — = ZiC 

Z] Z.2 z 2 



(5.74) 



Z, = 



n - 1 
L = Z, 



(5.75) 



where n = 1 + Zi C. 



228 FILTERS 



5.5.3 Design Example 

To design a low-pass microwave filter based on the transmission lines, let us choose a lumped third- 
order low-pass Chebyshev filter prototype shown in Figure 5.18(a) with the cutoff frequency f c — 
a> c /(27t) — 2 GHz, having a 0.5-dB equal-ripple response and an impedance R 0 — 50 Q. For unit 
element with R 0 — Zo — I and normalized inductance L3 = #3 = 1.5963, 

n = 1 + L/Z 0 = 2.5963. 

Using Table 5.3 and Eq. (5.31) results in the denormalized circuit parameters given by 

Lj = *!^ = l, = 6.375 nH 

C 2 = = 1.745pF. 

Q) C R 0 

Figure 5.21 shows the design transformation of a lumped low-pass filter prototype to a 
transmission-line filter using the Kuroda identities. The first step shown in Figure 5.21(a) is to 
add a unit element at the right end of the circuit and convert a series inductor into a shunt capacitor 
using the second Kuroda identity, as shown in Figure 5.21(b), resulting in the shunt capacitor having 
a normalized value of (n — \)ln — 0.6148. Then, adding another unit element at the left end of 
the circuit and applying the first Kuroda identity lead to the same result for the symmetrical filter 
structure, as shown in Figure 5.21(c) with resulting two unit elements and three shunt capacitors. To 
keep the same physical dimensions during the calculation of the circuit parameters, the inductance 
should be taken in nanohenri. The capacitance is measured in nanofarad if the cutoff frequency is 
measured in gigahertz. 

Finally, a Richards's transformation is used to convert the shunt capacitors to open-circuited 
stubs. In this case, the normalized characteristic impedance of an open-circuited stub equal to 1/Cj 
is necessary to be multiplied by Zq — 50 £2. Thus, the series transmission-line stubs have the 
characteristic impedance of 2.5963 x 50 = 129.3 Q, the left- and right-hand side open-circuited 
stubs have the characteristic impedance of 50/0.6148 = 81.3 £2 each, and the centre open-circuited 
stub has the characteristic impedance of 50/1.0967 = 45.6 Q. 

Figure 5.21(d) shows the transmission-line structure of the final low-pass Chebyshev filter. The 
lengths of the series and shunt open-circuited stubs are A/8 at 2 GHz. The simulated amplitude 
response of the designed transmission-line low-pass filter is shown in Figure 5.21(e), along with the 
response of its lumped-circuit prototype. It is clearly seen that the passband characteristics are very 
similar; however, the transmission-line filter has a sharper cutoff transition and its response repeats 
every 8 GHz as a result of the periodic nature of a Richards's transformation. 



5.6 COUPLED-LINE FILTERS 

The filter design can also be based on the parallel coupled transmission lines that can be easily 
implemented in microstrip or stripline form. First, let us consider the basic properties of a quarterwave 
line and a coupled-line section and then describe the most popular designs of the passband and 
stopband filter configurations. 

5.6.1 Impedance and Admittance Inverters 

The impedance and admittance inverters are useful to transform series-connected elements to shunt- 
connected elements, or vice versa. An ideal impedance inverter represents a two-port network shown 
in Figure 5.22(a) having a unique property at all frequencies which means that, if it is terminated in 



COUPLED-LINE FILTERS 229 




230 FILTERS 



Impedance inverters 



Admittance inverters 




(a) 



7*> = K 




o 1 



(b) 
V4 



y 0 =j 



o o 

FIGURE 5.22 Impedance and admittance inverters. 



an impedance Z L on one port considered a load, the input impedance Z in seen looking into the other 
port is 



7 — 



K 2 



(5.76) 



where A" is a constant image impedance of the inverter known as a A-inverter [19]. In this case, if 
Z L is capacitive (or inductive), then Z in will become inductive (or capacitive), and hence the inverter 
has a phase shift of ±90° or its odd multiple. The Ai?CZ)-matrix of the ideal impedance inverters can 
generally be written as 



A B 
C D 



0 



0 



(5.77) 



Similarly, an idealized admittance inverter is a two-port network shown in Figure 5.22(b) that 
exhibits such a unique property at all frequencies that, if an impedance Y L is connected at on one port 
considered a load, the input admittance Y in seen looking into the other port is 



Y,„ = 



,/ 2 



(5.78) 



where J is real and defined as the characteristic admittance of the inverter known as a ./-inverter. In this 
case, the admittance inverter has a phase shift of ±90° or its odd multiple as well. The ABCZ)-matrix 
of the ideal admittance inverters can generally be written as 



A B 
C D 



0 



±- 



1 

0 



(5.79) 



In its simplest form, a K- or 7-inverter can be implemented using a quarterwave transformer of 
the appropriate characteristic impedance, as shown in Figure 5.22(c). In this case, the ABCZ)-matrix 
of the inverter can be easily determined from the ABCD-parameters of the transmission line. 



COUPLED-LINE FILTERS 231 



5.6.2 Coupled-Line Section 

A parallel-coupled stripline section representing a four-port network with port voltages and currents is 
shown in Figure 5.23(a). It is well known that the even (symmetric) and odd (antisymmetric) excitation 
modes exist in a system of the two-coupled transmission lines. The transverse electromagnetic (TEM) 
even- and odd-mode electric field distributions in a homogeneous dielectric medium are shown in 
Figures 5.23(b) and 5.23(c), respectively. In this case, stronger coupling between the transmission 
lines takes place at an odd excitation mode, and Z 0e > Z 0 > Z 0o , where the characteristic impedance 
Zoe corresponds to even mode, the characteristic impedance Zq 0 corresponds to odd mode, and 
Z 0 = VZoeZk. is the characteristic impedance of a single transmission line. 

The four-port impedance Z-matrix for a lossless parallel coupled-line section is derived based on 
a superposition of the results at both types of excitation in a homogeneous medium, which can be 
written as 



-vr 




"Z„ 


v 2 






v 3 




Z31 


_v 4 _ 




_ Z41 



Z,2 

Z 22 
Z 3 2 
Z42 



Z13 
Z23 
Z33 
Z43 



Z14 
Z24 
Z34 
Z44 . 



h 
h 

Lh. 



(5.80) 



where 



Zn = Z22 = Z33 = Z44 = — j (Zoe + Zoo) 



Z12 = Z21 = Z34 = Z43 = —j (Zoe — Zoo) 



Z13 — Z31 — Z 2 4 — Z42 — — j (Z 0e — Z 0o ) 



Z14 = Z41 = Z23 = Z32 = — j (Zoe + Zoo) 



COt 6 

~Y 

cott 

~Y 
csc ^ 

CSC t 




(J) 

FIGURE 5.23 Parallel-coupled stripline four-port network, even- and odd-mode electric fields. 



232 FILTERS 




(</) 





0 












FIGURE 5.24 Equivalent circuits of coupled-line sections. 



where 6 is the electrical length of the transmission lines [20]. Equation (5.80) holds for a simplest 
case when both striplines have equal widths. However, by using unequal strip widths, an additional 
degree of freedom can be obtained as required in many practical cases [21]. 

Any two-port network with different boundary conditions can be built from the coupled-line 
section by terminating two of the four ports in either open or short circuits. In this case, the various 
circuits have different frequency responses, including low-pass, bandpass, all-pass, and all-stop. The 
most popular configuration of a bandpass filter can be realized with open-circuited conditions when 
1 2 — I 4 = 0, as open circuits are easier to fabricate than short circuits. From Table 5.1, the image 
impedance for the coupled-line section in terms of Z-parameters shown in the left-hand side of 
Figure 5.24(a) is 



Z n Z 2 l3 _ x/(Zoe - Z 0o ) 2 - (Zoe + Z 0o ) 2 COS 2 ( 

Z33 2 sin 6 



(5.81) 



When an electrical length of the coupled-line section is 6 — jt/2 (or it is k/4 long), the image 
impedance reduces to 



Z; = 



(5.82) 



which is real and positive since Z 0e > Z 0o - However, when 0 —> 0 or n, then Z ; —> 00, thus indicating 
a stopband, and the passband cutoff frequencies can be obtained from Eq. (5.81) when Z\ — 0 as 



cos Or. — ± - 



Zoe + Zoo 

The propagation constant can be defined using the results of Table 5.1 as 



(5.83) 



Zi 1Z3. 



Z()e + Zq, 



7- 



(5.84) 



which shows that fi is real for 0 C _ < 1 



c _, where 9 C _ and 6 C+ correspond to the low 



and high cutoff edges of the passband, respectively, and cos£* c _ = (Zo e — Zo 0 )/(Zo e + Zq 0 ) [20]. 



COUPLED-LINE FILTERS 233 



The design procedure for a transmission-line passband filter is based on an equivalence of a single 
coupled-line section and an equivalent circuit that includes an ideal impedance inverter with a phase 
shift of —90°, as shown in Figure 5.24(a). The ABCD-matrix of the entire equivalent circuit can 
be obtained from the multiplication of the ABCB-matrices of the ideal transmission lines and the 
impedance inverter as 



A B 
C D 



cos 6 jZ 0 smt 

smO 
; cos 6 



0 



(-- 


_ z ° 




" K 




sin 2 


; '(zl 



-j — 0 
J K 

/ 7 2 



K 



cos 9 j Z 0 sin t 
sin 6 



cos ( 



sin(9cose ; sin 1 6 - K cos 1 6 



K Zq . 

\ s'm6 cos 6 

Z 0 K 



(5.85) 



Then, the image impedance of the equivalent circuit can be found using Eq. (5.1) as 



Z 0 





Zo . 2 , 


K , 




— sin 6 


cos 2 9 




K 


Zo 




K . 






— sin 6 


" cos 2 6 


\ 


Zo 


K 



(5.86) 



while the propagation constant can be calculated from Eq. (5.4) as 



. K Zq . 

A = | I s'md cos ( 

. Zo K 



(5.87) 



By equating Eqs. (5.81) and (5.86) for the image impedances, Eqs. (5.84) and (5.87) for the 
propagation constants, and setting the electrical length of the coupled-line section to 6 — 90°, which 
corresponds to the center frequency of the bandpass response and for which a single-frequency equiv- 
alence is fully established, the separate equations for even- and odd-mode characteristic impedances 
can be derived as 



Zoe 


= 1 + 


Zo 




'Zo 










Zo" 


A' 




\K 


Zoo 




Zo 




'Zo 










Zo" 




A' 




V * 



(5.88) 
(5.89) 



Similarly, the equivalence between the coupled-line section with open-circuited and short-circuited 
conditions for a coupled line and the cascade of a unit element and a series inductor shown in Fig- 
ure 5.24(b) can be established [22]. In this case, the transmission-line low-pass filter can be fabricated 
using coupled-line sections with such a configuration where even- and odd-mode characteristic 
impedances can be derived as 

Z 0e = Z 0 + L + (Z 0 + L) (5.90) 

z 0o = — 2Z — (5 - 91) 

1 + — + — JL{Z Q + L) 

Zq Zq 



234 FILTERS 



where L is the series inductance of the equivalent circuit after application of Kuroda identities to the 
low-pass filter-prototype. Generally, there are ten possible combinations of boundary conditions for 
a coupled-line section [22]. 

In a practical implementation of the symmetric coupled stripline structure shown in Figure 5.23(d), 
the required strip widths and spacing for a given set of the characteristic impedances Zo e and Zq 0 and 
the dielectric constant e r for an idealized case of t — 0 can be calculated from 



94.15 



ki2 W 1 / nS 

+ — + — In 1 + tanh — 

n b 7t \ 2b 

94.15 



ln2 W 1 / jtS 

1 1 In ( 1 + coth — 

it b it \ 2b 



(5.92) 



(5.93) 



where W is the strip width, S is the strip spacing, and b is the distance between ground conductors 
[23]. Sufficiently accurate closed-form expressions for the effective dielectric constants and the 
characteristic impedances of coupled microstrip lines can be found in [24], 



5.6.3 Parallel-Coupled Bandpass Filters Using Half-Wavelength Resonators 

Narrowband bandpass filters can be designed using the cascaded coupled-line sections with open ends, 
as shown in Figure 5.25(a). In this case, the sections are of equal length (one-quarter wavelength 
at the center frequency), and their electrical design is completely specified by the even- and odd- 
mode characteristic impedances Zo e and Zq 0 , respectively. The total filter structure always will be 
symmetrical for maximally flat or equal-ripple response. In the design of a parallel-coupled with 
half-wavelength resonators, it is necessary first to select the type of response function and the number 
of resonators that will yield the desired transfer or attenuation function in the pass and stop bands. 
Then, the even- and odd-mode characteristic impedances for each coupled-line section are calculated 
based on Eqs. (5.88) and (5.89), respectively, from 



Z 0 



Zooi — Zq 



1 + ^ + 



(5.94) 



(5.95) 



where 



Zq I Aa> it 

Ki V 2a>o gi 
Zo Aa> 7t 

2a) 0 gi 
Zq I Aft) 7T 

#n+l V 2ft>0 £ngn+l 



for; = 2, 3, 



(5.96) 



gi, g2, ■ ■ ■ , gn+i are the normalized values for the ladder-type low-pass prototype elements from 
Tables 5.2 and 5.3, and coq — (a> c + + ft>c-)/2 is the center bandwidth frequency [8,19]. Finally, the 
transmission-line physical dimensions in each section should be designed to yield these characteristic 
impedances. In general, the strip widths and spacing will differ from section to section, and hence 
the width of the resonators will not be constant. The design equations have excellent accuracy for 



COUPLED-LINE FILTERS 235 




bandwidths up to 20% in the case of maximally flat response and 30% in the case of equal-ripple 
response. 

As an example, let us design a third-order (n — 3) coupled-line bandpass filter with the center 
frequency of 1 0 GHz and bandwidth of 20% , having a 0. 5-dB equal-ripple response and a characteristic 
impedance Zo = 50 Q. From Table 5.3, the element values for the low-pass prototype circuit are 
given by 

gi = 1.5963 g 2 = 1.0967 
g 3 = 1.5963 g 4 = 1.0000. 

Then, the expressions from Eq. (5.96) are used to determine the normalized impedance inverter 
constants in the form of Z 0 /A"j and finally the even- and odd-mode characteristic impedances Z 0e and 
Zoo can be calculated from Eq. (5.94) and (5.95), respectively, as 

Z oe = 82.02 n Zoo = 37.66 for/ = 1,4 
Z oe = 64.69 n Z 0o = 40.95 Q for/ = 2,3. 

The resulting amplitude response of the designed microwave third-order coupled-line passband 
filter in a frequency domain is shown in Figure 5.25(b), where the out-of-band suppression of 
40 dB is achieved below 6 GHz and above 14 GHz with a minimum suppression at 20 GHz. Note 



236 FILTERS 



that passbands also occur at higher frequencies, having the center bandwidth frequencies of 30 GHz, 
50 GHz, and so on. 



5.6.4 Interdigital, Combline, and Hairpin Bandpass Filters 

An interdigital bandpass filter represents an array of coupled-line resonators where each resonator 
element is a quarter-wavelength long at the midband frequency and is short-circuited at one end 
and open-circuited at the other end. A typical interdigital filter construction is realized by stripline 
suspending resonators in an air-filled metal case, and generally the physical dimensions of these 
resonators can be different. Coupling is achieved by way of the fields fringing between adjacent 
resonator elements. Coupled-stripline self- and mutual capacitances provide the starting point for 
the determination of the resonator width and spacing [25,26]. The mutual coupling between the 
resonators causes the resonator width to be less than the width of uncoupled lines. 

Figure 5.26(a) shows the interdigital bandpass filter structure with short-circuited lines at the 
ends where each element serves as a resonator, except for the input and output lines which have 



0 I 2 3 « « • n - 1 n n 



X 



X 



X 



K/4 



_L 



I 



* T 



I 2 3 4 ••»« — 2 n — 1 n 

„ T „ T _ X 



i 



I 



X 4 



T 



FIGURE 5.26 Interdigital passband filter structures. 



COUPLED-LINE FILTERS 237 



an impedance-matching function. This type of design is most practical for filters having narrow 
or moderate bandwidths. The main drawback in applying the design procedure to filters of wider 
bandwidths is that the gaps between lines 0 and 1 and between lines n and n + 1 tend to become 
inconveniently small when the bandwidth is large, and the thickness of lines 1 and n tend to become 
very small. The other interdigital filter structure is shown in Figure 5.26(£>) where the terminating 
lines are open-circuited that gives filter structural dimensions most suitable for moderate and wide 
bandwidths. In this case, all of the line elements serve as resonators. The main drawback of this filter 
structure for a narrowband design is that lines 1 and n will attain extremely high impedance. 

Interdigital passband filters have a number of attractive features: they are compact; the manu- 
facturing tolerances are relatively relaxed due to the relatively large spacing; the second passband 
is centered at three times the center frequency of the first passband, and there is no possibility of 
spurious responses in between; the rates of cutoff and the strength of the stopbands are enhanced 
by multiple-order poles of attenuation at dc and even multiples of the center frequency of the first 
passband [25]. In microstrip implementation, the interdigital filter is superior to the parallel coupled 
bandpass filter based on half-wavelength resonators because the microstrip interdigital filter occupies 
less space at the expense of shorts through the substrate [27]. However, its microstrip implementation 
suffers from severe asymmetry of the filter response due to effect of coupling between nonadjacent 
resonators. 

Figure 5.27 shows the tapped combline bandpass filter which consists of an array of coupled 
resonators that are short-circuited at one end and loaded by a grounded lumped capacitance C\ 
at the other end. In this case, the resonator lines will be less than a quarter-wavelength long at 
resonance each, and the coupling between resonators is predominantly magnetic [28]. However, if 
the capacitors C\ were not present, the resonator lines would be a full Ao/4 long at resonance, and the 
structure would have no passband. Without some kind of reactive loading at the ends of the resonator 
lines, the magnetic and electric coupling effects would cancel each other, and the combline structure 
would become an all-stop structure. The larger the loading capacitances, the shorter the resonator 
lines, which result in a more compact filter structure with a wider stopband between the first desired 
passband and the second unwanted passband. For example, when resonator length is equal to / = 
A 0 /8, then the second passband will appear at over four times the midband frequency, and when / = 
Ao/16, the second passband will be located at over eight times the midband frequency [29]. 

In practice, the minimum resonator line length is limited by the decrease of unloaded quality factor 
of resonator and a requirement of heavy capacitive loading. The bandwidth of combline filters is a 
function of the ground-plane spacing b and spacing 5 between resonators when the greater bandwidth 
is achieved for greater b and 5. The spacing b determines the resonator impedance and length, as well 
as the maximum power rating and quality factor. The loading capacitance for each resonator can be 



< 



2 


3 


4 . 


• • n - 2 


n- 1 


n 


± 


± 


I 


± 


± 


± 


I 


T 


T 


I 


T 


T 



z„ o- 



/< V4 



-O Zo 



Till 



III 



FIGURE 5.27 Structure of tapped combline passband filter. 



238 FILTERS 



2b O- 



-O Zo 



FIGURE 5.28 Structure of tapped hairpin-line passband filter. 



calculated as Q — cot6 0 /(a) 0 Z 0i ) where 0 0 — 2jtl/k 0 is the electrical length at the midband frequency 
and Zoi is the characteristic impedance of the zth resonator in view of the adjacent grounded lines i — 
1 and i + 1 . In the physical realization of narrow- and moderate-bandwidth interdigital and combline 
filters, the conventional input and output transformer couplings shown in Figure 5.26 are sometimes 
replaced with direct tapped connections shown in Figure 5.27 [30,31]. 

Hairpin-line passband filters, the basic tapped structure of which is shown in Figure 5.28, are 
preferred structure for microstrip realization because they offer small size and need no ground 
connection for resonators. This type of filters conceptually is obtained by folding the resonators 
of a parallel-coupled half-wavelength resonator filter into a U-shape. As a result, the same design 
equations for the parallel-coupled passband filter with half- wavelength resonators can be used [32]. 
However, to fold the resonators, it is necessary to take into account the reduction of the coupled-line 
lengths, which reduces the coupling between resonators. The line between two bends tends to shorten 
the physical length of the coupling sections, and the coupled section is slightly less than a quarter- 
wavelength [33]. Open-circuit resonators reduce free-space radiation due to phase cancellation of 
fields at the ends. Also, the radiation decreases with decreasing space between the folded lines of 
the hairpin. However, when this space is small, self-resonator coupling causes a decrease in filter 
bandwidth and an increase in losses. A reasonable spacing is two to three times the inter-resonator 
spacing, or five times the substrate thickness [34]. Microstrip hairpin filters require a sufficiently large 
spacing between resonators to achieve the desired narrow passband. 

The hairpin-line bandpass filter design parameters can be calculated from the low-pass filter 
prototype parameters g; by 



Gel 



gogi 
Aw 



TT Aft) 

C u+ i = -— = fori = 1,2,..., n-1 (5.97) 



2e„ = 



gngn+1 

Aw 



where Am is the passband, <2 e i and Q tD are the external quality factors of the resonators at the input 
and output, and Cj. ; + i are the coupling coefficients between the adjacent resonators [11]. 

The input and output resonators can be slightly shortened to compensate for the effect of the 
tapping line and the adjacent coupled resonator. The design equation for estimating the tapping 
length t can be written as 



21 . _j / n Z 0 



t= ~ sin J^TT^r (5.98) 

^Ue ^0i 



where Z a is the characteristic impedance of the hairpin line, Z 0 is the terminating impedance, 
and I is the length of the coupled section [35]. Although the design equation ignores the effects of 



COUPLED-LINE FILTERS 239 



discontinuity at the tapped point and coupling between two folded arms, such estimation is sufficiently 
accurate. In the microstrip hairpin-line bandpass filter, a spurious mode occurs at approximately twice 
the passband frequency due to the different even- and odd-mode propagation velocities of coupled 
resonators. To resolve this problem, the tapped length t can be chosen close to X 0 /S. In addition, 
to minimize the overall size and make microstrip bandpass hairpin-line filters more compact, their 
structures in the cross-coupled or zigzag form can be used [36,37]. 



5.6.5 Microstrip Filters with Unequal Phase Velocities 

The conventional approach to design parallel-coupled microstrip passband filters results in the spuri- 
ous response at twice the passband frequency, which causes passband response to be asymmetric and 
reduces the width of the upper stopband [38]. This is a result of the inequality of the even- and odd- 
mode phase constants of the coupled line for each stage, which becomes more severe with high relative 
permittivity of the dielectric substrate. In this case, to partially solve this problem, the different lengths 
for the even- and odd-modes or overcoupling of the end stages with high image impedance of the filter 
can be used [39,40], The lumped capacitors can be connected to the resonators to extend the traveling 
path of the odd mode [41,42]. Besides, the optimization of the resonator length and characteristic 
impedances can improve the return loss in the passband and out-of-band rejection [43,44], 

However, the effect of unequal even and odd modes with high velocity ratios can alternatively 
be used to introduce some novel properties of the filters based on an inhomogeneous coupled-line 
structure. For example, a single microwave type-C filter section based on inhomogeneous broadside- 
coupled strips with phase velocity ratio of 2, which is an all-pass circuit in the homogeneous case, 
achieves an equal-ripple 3-peak stopband and passband response [45,46]. 

The TEM even- and odd-mode electric field distributions in an inhomogeneous dielectric medium 
with relative permittivity e r are shown in Figure 5.29(a), where the stronger coupling between the 
transmission lines takes place at an odd excitation mode. For each excitation mode, the characteristic 
impedance Z 0e corresponds to even mode, while the characteristic impedance Z 0o corresponds to odd 
mode. In this case, 

Z 0e > Z 0 > Z 0o (5.99) 

where Zq — V-Z)eZo 0 is the characteristic impedance of a single transmission line. The equivalent 
representation of a single open-circuit comb-line section can be given by a lattice filter section with 
semilattice impedances Z a and Z b , as shown in Figure 5.29(£>), where Z a is the impedance of the 
equivalent two-port network defined as the input impedance of the original two-port network under 
its odd-mode excitation, while Z b is the corresponding impedance under even-mode excitation. In 
this case, an independent control of the odd mode only, resulting in high phase velocity ratio, can be 
provided by adjusting the position of the short circuit between the conductors along the coupled-line 
section, as shown in Figure 5.29(c), with no odd-mode field existing between the short circuit and the 
open ends. The overall result is defined by a superposition of the results at both types of excitation. 

The four-port impedance Z-matrix for a single comb-line section shown in Figure 5.29(b) in an 
inhomogeneous dielectric medium can be written as 



[Z]=i 
2 



Z D + z a z b — z a 
z b — z a z b + z a 



(5.100) 



where 



Z a = ;X a = -./Z 0o cot(^-^) (5.101) 

JOo 
2 / 0 e 



Z b = jX h = -yZoeCot l^-4-\ ( 5 - 102 ) 



240 FILTERS 




FIGURE 5.29 Parallel-coupled microstrip structure with even and odd electric field modes, its equivalent 
representation, and frequency response. 

where / 0e and f 0o are the frequencies at which a comb-line section are a quarter- wavelength long 
electrically when excited in the even- and odd mode, respectively [47,48]. 

The effective attenuation of a comb-line section in terms of the equivalent lattice circuit parameters 
can be written as 

/l + X'X'\ 2 

where X' a and X' b represent the corresponding reactances X a and X^ normalized to Zo [4,7]. 
The image impedance can be obtained from Eq. (5.5) as 



Z; = v Z a Zb = j v X^Xb 



(5.104) 



COUPLED-LINE FILTERS 241 



From Eqs. (5.103) and (5.104) it follows that, in a inhomogeneous dielectric medium with unequal 
phase velocities when/ 0e 7^/00. the image impedance Z; of a comb-line section becomes real in a 
frequency region with boundary frequencies /o e and/o 0 , as shown in Figure 5.29(d). This is because 
the reactances X. d and X b have different signs in this region, negative for X a and positive for X b , 
with zeros of the attenuation function A defined by Zi(/o) = Zq [49,50]. In a homogeneous dielectric 
medium, the electrical lengths for even and odd modes are equal, and a comb-line section represents 
an all-stop circuit when the frequency region of real image impedances degenerates to / 0e = fo 0 — 
foo, with an attenuation pole at this frequency. 



5.6.6 Bandpass and Bandstop Filters Using Quarter- Wavelength Resonators 

The bandpass or bandstop response can be achieved by using quarter-wavelength open- or short- 
circuited transmission-line stubs which can be placed in shunt along a transmission line, as shown 
in Figure 5.30 for 0 — nil. In this case, each open-circuit stub acts as a series resonant circuit, 
whereas each short-circuit stub operates as a parallel resonant circuit. Quarter-wavelength sections 
of the transmission line between stubs can be considered as impedance inverters to convert alternate 
shunt resonators to series resonators. For narrow bandwidths, the characteristic impedance of the 
series quarter-wavelength transmission lines is equal to Z 0 . Bandstop filters can be used to eliminate 
a band of frequencies, or to separate such a band from other frequencies. 

The design equations for the required stub characteristic impedances Z 0 „ is derived based on 
the element values of a low-pass filter prototype. Generally, the input impedance of an open-circuit 
stub with the characteristic impedance Zo and electrical length 8, which can be approximated as 
a series LC resonator shown in Figure 5.31(a), can be written as Z in = — jZ 0 cot#, whereas the 
input impedance of a short-circuit stub with the characteristic impedance Z 0 and electrical length 6, 
which can be approximated as a parallel LC resonator shown in Figure 5.31(h), can be written as 





FIGURE 5.30 Bandpass and bandstop filters with shunt quarterwave resonators. 



242 FILTERS 




Z; n = jZo tanO. As a result, the characteristic impedance Zoi of each open-circuit transmission-line 
stub used in the n-order bandstop filters is calculated from 

4Znft>0 

ngiAa) 

whereas the characteristic impedance Zo; of each short-circuit transmission-line stub is determined 
by 

Z 0i = — ? (5.106) 



where g ; are the element values for maximally flat or equal-ripple low-pass filters prototypes, Aa> is 
the frequency bandwidth, and a> 0 is the center bandwidth frequency, i— 1, 2, ... ,n [5,51]. Note that 
these results can only be applied to filters having input and output impedances of Zo, and cannot be 
used for equal-ripple designs with even n. 

As an example, let us design a third-order (n = 3) coupled-line bandstop filter with the center 
frequency of 4 GHz and bandwidth of 25%, having a 0.5 dB equal-ripple response and a characteristic 
impedance Z 0 — 50 Q. For the corresponding element value of the low-pass prototype circuit tabulated 
in Table 5.3, the characteristic impedance Z oi is calculated as 

gi = 1.5963 Zoi = 159.5 Q fori = 1 
g 2 = 1.0967 Z 02 = 232.2 Q fori = 2 
g 3 = 1.5963 Z 03 = 159.5 fori = 3 



resulting in the transmission-line filter topology shown in Figure 5.31(c) and amplitude response 
shown in Figure 5.31(d). In this case, the ripple in the passband is somewhat greater than expected, 
which is the result of a narrowband assumption corresponding to the stop-band bandwidths up to a 
few percent. 



SAW AND BAW FILTERS 



243 



For better accuracy over wider bandwidths, the characteristic impedances of the interconnecting 
quarter- wavelength lines must be variable calculated according to the corresponding design equations 
[8,52]. These distributed microwave bandstop filters are characterized by their second harmonic 
response centered at no more than three times the fundamental bandstop center frequency. However, 
such a limitation can be removed by the use of compound stub resonators, each composed of 
a pair of transmission lines with different characteristic impedances with each line being of the 
commensurate length when the first upper stopband center frequency becomes as high as six times 
the fundamental bandstop center frequency [53]. By providing a parallel association of two different 
bandstop structures, the narrow bandpass filters with short- and open-circuit stubs of different lengths 
and characteristic impedances can be realized which allow an independent control of each attenuated 
band on either side of one bandpass [54]. 

5.7 SAW AND BAW FILTERS 

Serious interest in surface acoustic wave (SAW) technology commenced in 1967 with the develop- 
ment of the interdigital electrode transducer (IDT), which permitted efficient transduction between 
electromagnetic acoustic energy [55]. The IDT consists of a set of interleaved metal electrodes and 
is fabricated by photoetching a deposited metal film. In the simplest form, the width and spacing 
of electrodes is equal and uniform throughout the pattern. In this case, electrical excitation of the 
transducer with a sinusoidal voltage produces a periodic electrical field which penetrates into the 
piezoelectric substrate. In this case, suitable choice of substrate orientation produces two surface 
acoustic wave beams propagating normal to the IDT electrodes. As a result, peak output occurs 
at the synchronous frequency f 0 — v/l, where v is the SAW velocity and / is the transducer peri- 
odic length. Under this condition of acoustic synchronism, the stress contributions of all electrodes 
add in phase. 

The lower SAW filter frequency limit is set by practical size limits on piezoelectric substrates 
and the range of velocities of available surface wave materials. At the high-frequency end, the limit 
is set by the resolution capabilities of state-of-the-art patterning techniques, substrate velocities, and 
frequency-dependent loss mechanisms. The essential feature in changing the operating frequency of 
surface-wave filters is simply a matter of changing the interdigital electrode spacing. In this case, 
the width of acoustic beam is set by electrode stripe length w which can be adjusted to obtain a 
convenient impedance level. With increasing frequency, however, electrode thickness must decrease 
to avoid larger distortion from reflections due to increased parasitic resistive loss. Several filter 
applications were found at both ends of available surface-wave frequency range, from the frequency 
modulated (FM) radio filters at 10.7 MHz with a 200-kHz signal bandwidth to the 800-MHz filters 
for mobile telephone transceivers and L-band radar and satellite transponder filters operating at 1.5 
GHz and higher [56-58]. To achieve higher SAW operating frequency, it is necessary either to use 
substrate with higher velocities than can be achieved in quartz and lithium niobate, or to develop 
higher resolution fabrication. An important relationship in SAW filter design is that between fractional 
bandwidth (Af/fo) and insertion loss, with benefit of lithium niobate substrate over quartz substrate 
due to stronger piezoelectric coupling despite its moderate temperature sensitivity. The out-of-band 
rejection better than 60 dB over a wide range of bandwidths is available with weighting techniques 
such as finger overlap, finger withdrawal, phase weighting, and multistrip couplers. Principle obstacles 
to achieve or improve such a performance include diffraction and spurious responses. 

Figure 5.32 shows the equivalent electric circuit for IDT, where G represents the radiation con- 
ductance, Ct represents the capacitance between transducer fingers, and B represents the acoustic 
susceptance due to vibration [59]. This circuit corresponds to a shunt representation for transducer 
input immittance in a "crossed-field" model configuration when the applied electric field is normal to 
the acoustic propagation vector. However, it can equivalently be illustrated by a series representation 
for transducer input immittance in an "in-line" model configuration characterized by parallel electric 
field and propagation vectors. For a given piezoelectric with given cut and orientation, the choice 



244 FILTERS 




FIGURE 5.32 Equivalent electric circuit for IDT. 



between these models is made by evaluating the contribution to Rayleigh wave (RW) excitation of 
the perpendicular and parallel to the surface electric field components. 

From computer simulation and practical experiments, it was found that the input impedance 
(or admittance) of an IDT with quarter-wavelength-solid fingers can be classified into three types, 
depending on the number of finger pairs N and electromechanical coupling coefficient defined as 
k 2 — 2Av/v, where Ad is the perturbation of phase velocity due to nonidentical field distributions for 
interdigital and metal-coated configurations [60]. In this case, for an optimized N — l.5/k 2 , a wide 
frequency range can be achieved. The total susceptance (coCj + B) becomes very small over this 
range due to the cancellation of electric and acoustic susceptance components, being capacitive for 
smaller N and inductive for larger N. 

Figure 5.33 shows the configuration with sharp cutoff frequency response representing an image- 
impedance connection where one pair of electrically connected interdigital transducers with an 
optimum number of finger pairs is introduced. Input and output transducers with a broader frequency 
response are arranged on both sides where SAW reflectors are also placed [61]. With this nonweighting 
configuration, a sharp cutoff response is achieved because, in the passband where the susceptance is 
cancelled, the current from an upper image-impedance IDT flows into a lower one without reflection. 
If the frequency is outside passband, almost all current is reflected at the image-impedance connected 
point because the susceptance component is larger than the conductance component. In this case, 
an insertion loss of as low as 3.5^.0 dB is achieved for the ultra high frequency (UHF) filter, and 
1.7-2.0 dB for the very high frequency (VHF) filter. 

The loss mechanism in a UHF SAW filter is basically defined by the leakage (bidirectionality), 
weighting, and thin Al electrode conductivity losses. Since bidirectionality loss is the largest in 
conventional transversal SAW filters, an introduction of the resonant structure with lateral repetitions 
in addition to reflectors can minimize this loss. To reduce weighting loss, a new phase weighting, 
which excites SAW structure with uniform wavefront distribution in the passband, can be included. 
Figure 5.34 shows the new repetition structure for interdigital transducers with a tapering number of 
finger pairs where one pair of input and output transducers is laterally repeated [61]. Since there is a 
larger number of finger pairs in the center of the filter and a smaller number at both sides, thus the 
energy of acoustic vibrations is confined within the filter. As a result, six repetitions are enough to 
achieve 0.7 dB of insertion loss using a simple taper in the number of finger pairs (half the number 
of finger pairs at both sides). To reduce the conductivity loss, the filter aperture should be small and 
the impedance of each repeated IDT must be increased. 

The device will usually be hermetically packaged to protect the sensitive surface from contam- 
ination. Often, one or two reactive components must be added at each end, outside the package. 
An inductor may be needed because the interdigital transducers have capacitive reactance which 



SAW AND BAW FILTERS 



245 



In 
1 




Out 



FIGURE 5.33 SAW filter with sharp cutoff frequency response. 




FIGURE 5.34 SAW filter with tapered number of finger pairs. 



246 FILTERS 



dB 




736.5 786.5 836.5 886 5 936.5 

Frequency, MHz 

FIGURE 5.35 Typical SAW filter response. 



may need to be tuned out. Also, lumped LC circuits are often used to transform the source or load 
impedance (usually 50 Q) to an impedance more suitable for the SAW device. Typical SAW filter 
response for an 800-MHz mobile telephone transceiver is shown in Figure 5.35. To improve its 
power capability, insertion loss, and out-of-band rejection satisfying the requirements for a miniature 
antenna duplexer, the transmitter SAW filter can be realized in a ladder-type configuration with addi- 
tional shunt capacitors between electrode fingers and ground [62]. Extremely low-loss and ultra-steep 
cutoff SAW filters used in a 1 .9-GHz antenna duplexer can be achieved with IDT metallization ratio 
less than 50% and optimized ladder-type structure with a reduction in the coupling coefficient k 2 for 
only selected series resonators [63,64]. To minimize SAW filter size for multimode mobile phone 
applications, a chip-sized SAW package, which is flip-chip mounted onto a chip carrier serving as 
bottom of the package with electrical connections to the chip provided by solder bumps, represents a 
key packaging technology [65]. 

The traditional SAW filter technology demonstrates good selectivity and very small size, but 
operation frequencies of SAW filters is limited below 3 GHz due to their high insertion losses at 
high input power levels of above 1 W. On the other hand, despite the recent advances in ceramic 
material technology which have made dielectric filters very largely employed as duplexers since they 
can handle high power levels and demonstrate high selectivity, their large dimensions limit their 
application in mobile RP front-end modules. At the same time, the radio-frequency bulk acoustic 
wave (BAW) filters and duplexers based on film bulk acoustic resonator (FBAR) filters or solidly 
mounted resonator (SMR) technologies are expected to replace these traditional RF filter technologies 
due to their lower insertion loss, better selectivity, higher power handling, higher operation frequency. 
Besides, they can be directly integrated with active circuits, making possible of fully-integrated RF 
front-ends at very competitive costs. 

The core element for both BAW and SAW filters technologies is the resonator which is char- 
acterized by high quality factor due to low propagation loss and small size due to much shorter 
acoustic wavelength than that of the electromagnetic wave. Besides, acoustic waves can be easily and 
efficiently excited and converted back to electrical signal by using transducers and the piezoelectric 
property of the SAW and BAW materials. Though the impedance responses of these filters are similar, 
the resonances are realized quite differently in SAW and BAW filter technologies. In the former case, 
a surface acoustic wave is an elastic wave that travels along the surface of a crystal substrate, however 



SAW AND BAW FILTERS 



247 




R, 



-VW 




h h 

(e) 

FIGURE 5.36 BAW resonator structure, its equivalent circuit, and frequency response. 



in the latter case, a bulk acoustic wave represents an elastic wave that travels inside solid material. In a 
SAW resonator, the wave amplitude decays rapidly with depth, with 90% or more energy of acoustic 
energy being within one wavelength of the surface material. In a BAW resonator whose structure 
is shown in Figure 5.36(a) together with a signal source, the acoustic wave which is called the 
longitudinal wave propagates in a vertical direction between top and bottom metal electrodes [66,67]. 

Generally, the equivalent circuits of BAW and SAW resonators valid near the fundamental reso- 
nance are the same and are shown in Figure 5.36(b), where Co is the static (or plate) capacitance, C m is 
the motional (or acoustic) capacitance which is much smaller than Co, L m is the motional inductance, 
R m is the motional resistance, and the resistance R s and Rq characterize the electrode ohmic loss and 
undesired acoustic wave loss, respectively [68,69]. Being a second-order circuit, it has both a zero 
which defines the series resonant frequency f s and pole which defines the parallel resonant frequency 
/ p , as shown in Figure 5.36(c). In this case, the motional inductance forms a series resonance with 
C m and, at a slightly higher frequency, a parallel resonance with Co, with the inductive impedance in 
between f s and/ p . 

The most important characteristic of a BAW resonator is the strength of the piezoelectric coupling 
which depends on the piezoelectric and electrode materials and the resonator structure. With a single 



248 FILTERS 



piezoelectric plate without any layers, the coupling coefficient can be determined from the series and 
parallel resonance frequencies as 

n f* .„„ ( 71 U~ /« 



'■■jniYJ ,5 - 107> 

However, in a real FBAR resonator, there are other layers and, then, the effective piezoelectric 
coupling coefficient is defined by 



\/p/ C m + Co 7T Z 

which means that one must bear in mind that k — 1 . lfc eff when comparing these coupling coefficients 
[70]. For a BAW resonator, the only piezoelectric thin-film material that has been established as the 
best balance of performance and manufacturability is aluminum nitride (AIN). With the best layer 
stack design and with optimized AIN thin-film deposition system, the maximum achievable resonator 
relative bandwidth BW = (f p -/ s )// p and k 2 tSS are 2.8% and 6.9%, respectively [66]. 

The BAW (or SAW) resonator filters can be generally divided in two basic topologies: ladder shown 
in Figure 5.37(a) and lattice shown in Figure 531(b), or the combination of them [71,72]. Ladder-type 
BAW filters have a very steep rejection response at passband ends followed by a lower rejection at 
further frequencies. On the other hand, lattice-type BAW filters demonstrate slower roll-off coefficient 
and higher rejection at both stopbands which are more effective to for rejecting undesired frequency 
bands. As a result, a combined ladder-lattice BAW filters can provide high in-band selectivity and 
improved out-of-band isolation simultaneously. Using FBAR technology with acoustic loading layer, 




FIGURE 5.37 Ladder and lattice BAW filter configurations. 



SAW AND BAW FILTERS 



249 




dB 






(/') 








(< 


) 


1 



FIGURE 5.38 Operation principle of filter L-section. 



the double lattice BAW filter can achieve an insertion loss of 2.5 dB, a bandwidth of 250 MHz, and 
selectivity around 56 atX-band (11.325 GHz) [73]. 

The operation principle of a ladder-type filter can be explained based on its basic L-section which 
is composed of the consecutive series and shunt resonators, as shown in Figure 5.38(a). The series 
resonator is characterized by the series resonant frequency f sl and parallel resonant frequency f pl . At 
the same time, the shunt resonator is tuned to operate at a slightly lower frequency by increasing the 
period of the interdigital transducer for SAW resonator or by mass loading the electrode for BAW 
resonator. When its parallel resonant frequency / P 2 is chosen to be equal to or slightly lower than 
its series resonant frequency / s i , as shown in Figure 5.38(£>), a passband is formed at frequencies 
around /p 2 and/ sl , as shown in Figure 5.38(c), where there is a low impedance between the input 
and output terminals of the L-section and a high impedance between the output terminal and ground. 
At frequency / S 2, any current flowing into the L-section is shorted to ground by the low impedance 
of the shunt resonator, and a zero is seen in the transition response below the passband. At frequency 
fpi, the impedance between the input and output terminals of the L-section is high and another zero 
occurs in the transmission response above the passband. Far below / S 2 and above f p \, the L-section 



250 FILTERS 



S 2 u dB 











































52 dB 

































1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 

Frequency, GHz 
FIGURE 5.39 Frequency response of antenna BAW duplexer. 



behaves as a two-capacitor network characterizing by insufficient out-of-band rejection. However, 
better rejection can be achieved by cascading more L-sections and by trading off with insertion loss. 

The typical frequency response measured in the antenna BAW duplexer is shown in Figure 5.39, 
with an insertion loss in the passbands of 2.5 dB for transmitter path to the antenna and 3.0 dB 
for receiver path. In the path from transmitter to antenna, the insertion loss within the passband is 
definitely not the limiting factor because that role is played by the excessive roll-off at the filter edges. 
The total attenuation level in the signal path from antenna to receiver is greater due to the additional 
half-stage in the ladder-type topology of the receive filter. Typical suppression of the receive signal 
in the transmit band is 52 dB, that of the transmit signal in the receive band is 48 dB. 



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Stepped-Impedance Stubs and Different-Length Stubs," IEEE Trans. Microwave Theory Tech., vol. MTT-52, 
pp. 1034-1044, Mar. 2004. 

55. P. M. Grant, J. D. Adam, and J. H. Collins, "Surface Wave Device Applications in Microwave Communication 
Systems," IEEE Trans. Commun., vol. COM-22, pp. 1410-1419, Sept. 1974. 

56. R. D. Hays and C. S. Hartmann, "Surface-Acoustic-Wave Devices for Communications," Proc. IEEE, vol. 
64, pp. 652-671, May 1976. 

57. Y. Kinoshita, M. Hikita, M. Toya, T. Toyama, and Y. Fujiwara, "Low Loss and High Performance SAW Filters 
for 800 MHz Mobile Telephone Transceiver," IEEE Trans. Vehicular Technol, vol. VT-32, pp. 225-229, Aug. 
1983. 

58. H. Shinonaga and Y. Ito, "Microwave SAW Bandpass Filters for Spacecraft Applications," IEEE Trans. 
Microwave Theory Tech., vol. MTT-40, pp. 1110-1116, June 1992. 

59. W. R. Smith, H. M. Gerard, J. H. Collins, T. M. Reeder, and H. J. Shaw, "Analysis of Interdigital Surface Wave 
Transducers by Use of an Equivalent Circuit Model," IEEE Trans. Microwave Theory Tech., vol. MTT-17, 
pp. 856-864, Nov. 1969. 

60. M. Hikita, H. Kojima, T. Tabuchi, and Y. Kinoshita, "800-MHz High-Performance SAW Filter Using 
New Resonant Configuration," IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 510-518, June 
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61. M. Hikita, T. Tabuchi, and A. Sumioka, "Miniaturized SAW Devices for Radio Communication Transceivers," 
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62. M. Hikita, Y. Ishida, T. Tabuchi, and K. Kurosawa, "Miniature SAW Antenna Duplexer for 800-MHz 
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pp. 1047-1056, June 1988. 

63. J. Tsutsumi, S. Inoue, Y. Iwamoto, T. Matsuda, M. Miura, Y. Satoh, M. Ueda, and O. Ikata, "Extremely 
Low-Loss SAW Filter and Its Application to Antenna Duplexer for the 1 .9 GHz PCS Full-Band," Proc. IEEE 
Int. Frequency Control Symp., pp. 861-867, May 2003. 

64. S. Inoue, J. Tsutsumi, T. Matsuda, M. Ueda, O. Ikata, and Y. Satoh, "Ultra-Steep Cut-Off Double Mode SAW 
Filter and Its Application to a PCS Duplexer," IEEE Trans. Ultrasonic Ferroelectric Frequency Control, vol. 
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65. H. Meier, T. Baier, and G. Riha, "Miniaturization and Advanced Functionalities of SAW Devices," IEEE 
Trans. Microwave Theory Tech., vol. MTT-49, pp. 743-748, Apr. 2001. 

66. F. Z. Bi and B. P. Barber, "Bulk Acoustic Wave RF Technology," IEEE Microwave Mag., vol. 9, pp. 65-80, 
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67. M. Ueda, M. Iwaki, T. Nishihara, Y. Satoh, and K. Hashimoto, "A Circuit Model for Nonlinear Simulation 
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68. J. F. Rosenbaum, Bulk Acoustic Wave Theory and Devices, London: Artech House, 1988. 

69. M. Ylilammi, J. Ella, M. Partanen, and J. Kaitila, "Thin Film Bulk Acoustic Wave Filter," IEEE Trans. 
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70. H. Jaffe and D. A. Berlincourt, "Piezoelectric transducer materials," Proc. IEEE, vol. 53, pp. 1372-1386, Oct. 
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73. A. A. Shirakawa, J.-M. Pham, P. Jarry, and E. Kerherve, "Design of FBAR Filters at High Frequency Bands," 
Int. J. RF and Microwave Computer- Aided Eng., vol. 17, pp. 115-122, Jan. 2007. 



Modulation and Modulators 



Modulation is the process that allows the information content of an audio, video, or data signal to 
be transferred to a region of higher frequencies where the transmission can be effective. In this case, 
the spectrum bandwidth of the modulating signal must be much smaller compared to the carrier 
frequency being modulated. During the modulation process, one or several parameters of the carrier 
signal such as amplitude, frequency, or phase vary according to the modulating signal when the 
modulator changes the signal into a form suitable for transmission over the proper radio channel. 
More complex modulation process includes digitization and encoding of the modulating signals. 
Depending on the communication systems with corresponding requirements on transmitting power, 
signal quality, frequency bandwidth, power consumption, system complexity, or cost, different types 
of the modulation scheme can be used based on analog and digital modulation techniques. 

This chapter discusses the basic features of different types of analog modulation including ampli- 
tude, single-sideband, frequency, and phase modulation and basic types of digital modulation such 
as amplitude shift keying, frequency shift keying, phase shift keying, or pulse code modulation and 
their variations. The principle of operations and various schematics of the modulators for different 
modulation schemes including Class-S modulator for pulse-width modulation are described. Finally, 
the concept of time and frequency division multiplexing is introduced, as well as brief description of 
different multiple access techniques is given. 

6.1 AMPLITUDE MODULATION 

The amplitude modulation (AM) concept invented in 1902 by Reginald Fessenden and followed by 
his audio radio broadcast demonstrations in 1906 by means of the alternator-transmitter became the 
original method used for audio radio transmissions [1]. The basic design idea was to produce a steady 
radio signal when connected to an aerial by simply placing a carbon microphone in the transmission 
line, and the strength of the radio signal could be varied in order to add sounds to the transmission. 
This means that AM would be used to impress audio on the radio frequency earner wave. However, 
it would take many years of expensive development before even a prototype alternator-transmitter 
would be ready, and a few more years beyond that for high-power versions to become available. Later 
in 1920s, audio radio broadcasting became widespread by using the vacuum-tube transmitters rather 
than the alternator. 

6.1.1 Basic Principle 

The basic single-frequency carrier signal v — Vcosfwof + 4>o) is characterized by the constant 
amplitude V, carrier frequency cd 0 , and initial phase <j> 0 that can be set to zero. Such a signal does 
not contain any information since it repeats over and over again with the same parameters. However, 
when this signal is modulated, either in amplitude or frequency, it is no longer a simple sine wave, but 
is instead a mixture of several waves of slightly different frequencies, superimposed upon each other. 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



255 



256 MODULATION AND MODULATORS 



Thus, in the common case of modulating signal v m , the AM signal with time-varying amplitude V = 
V 0 + v m (t) can be written as 

v(t) = [V 0 + t) m (f)]costuo* (6.1) 

where Vq represents the carrier amplitude without modulation. Generally, such a transmitting signal 
may represent any periodical and non-periodical processes, the spectral components of which occupy 
very wide frequency range. However, the main signal energy is concentrated in a sufficiently narrow 
frequency bandwidth that gives an opportunity to make the communication channel bandwidth much 
narrower to transmit signal with required quality characterized by acceptable distortion level. 

In a simple case of a sinusoidal modulating signal v m (t) — V m cosfif, where V m is the modulating 
amplitude and Q is the modulating frequency, Eq. (6.1) can be rewritten as 

v(t) — Vq [1 + m cos Qt] cos coot (6.2) 

where m is the modulation index or modulation factor defined as 

m = ^ (6.3) 

In this case, the maximum and minimum values of the AM signal amplitude is obtained from 
Eq. (6.2) as V max = Vq(1 + m ) and ^min = Vo(l — ni), respectively, and modulation factor can be 
characterized through the relative amplitude variation as 



' nun 



(6.4) 

V 4- V - 

v max v mm 

as shown in Figure 6.1(a). 

Equation (6.2) can be rearranged by employing a trigonometric identity into the form 

m V 0 m Vq 
v(t) = Vocosa> 0 f H — cos (two + £2) t H — cos (co Q — Q)t (6.5) 

where the first term is called the carrier, which is unaffected by modulation process and the amplitude 
of which is equal to Vq. The other terms at frequencies a>o + Q and a>o — f2 are called the upper 
and lower sideband frequencies, respectively. Their amplitudes are proportional to the modulation 
factor and their frequencies differ from the carrier frequency by an amount equal to the modulating 
frequency. Figure 6.1(b) shows the frequency spectrum of such a single-frequency AM signal with 
a bandwidth of 2f2. The relative phase relationships of these three components are illustrated in 
Figure 6.1(c), with the vector of the carrier frequency rotating counterclockwise in a circle with 
angular velocity coq and the side-frequency vectors rotating with angular velocities coq + f2 and 
coo — Q, respectively. Thus, the side-frequency vectors are rotating in opposite directions relatively 
to the carrier vector with the angular velocity Q, and their vector sum will always lie along the carrier 
vector, indicating amplitude variation but no frequency deviation. This means that, in the complex 
plane, both the carrier and the sum of the side-frequency phasors are always located on the real 
horizontal axis. Hence, in AM, carrier and modulation vectors are in phase [2]. Consequently, the 
resulting AM-signal vector is rotating with the constant angular velocity &>o, and its length varies 
periodically with the angular velocity Q. 

Figure 6.2 shows the time-domain behavior of the AM signal for fixed values of carrier and 
modulating frequencies and different modulation factors: (a) m — 50%, (b) m — 100%, and (c) m — 
150%. In this case, for m < 1, the carrier time- varying amplitude or envelope of the modulated wave 
varies in proportion to the time- varying modulating signal v m (t) — V m cos£2f. However, for m > 1, 
the modulation symmetry is violated since the negative peaks of v m when V m > Vo reduce the total 



AMPLITUDE MODULATION 257 




voltage to the points less than zero. In this case, the resulting AM signal envelope do not corresponds 
anymore to the modulating signal waveform. This condition is called the overmodulation, resulting 
in a significant nonlinear distortion for the transmitting signal. 

In a common case of the complicated modulating signal shown in Figure 6.3, the upper and 
lower frequency sidebands appear, each of which corresponds to the modulating signal spectrum. If 
^max is the highest modulating angular frequency present and co 0 is the carrier angular frequency, 
the complete spectrum of the amplitude-modulated waveform extends from the bottom of the lower 
sideband at cdq — £! max to the top of the upper sideband coq + f2 max , with a total bandwidth of (&>o + 

^max) (^0 ^max) = 2£2 ma x- 




258 MODULATION AND MODULATORS 



I 



W 0 - fin, 



2n„ 



FIGURE 6.3 Amplitude modulation spectrum representation. 



The carrier power without modulation can be written as 

P 0 = *L (6.6) 

where R L is the load resistance. At the same time, the maximum power P max corresponding to the 
maximum AM signal amplitude V max is defined as 

^ = ^ = ^^ = 'o(lW (6.7) 

which is also called the peak envelope power (PEP). 

From Eq. (6.7) it follows that, for m — 1 when maximum possible linear transmission mode 
without overmodulation can be realized, the maximum power P max (or peak envelope power Ppep) 
is four times greater than the carrier power Pq, that is, P msx — 4Pq. The average power P ml over 
complete cycle of the modulating signal can be found as 

2jr 

2 ■> 



= ^ j P 0 (l + m cos Qtf d (tit) = P 0 ^1 + ( 6 - 8 ) 



where the total sideband power is equal to P sb — (m 2 /2)P 0 . In this case, the bandwidth of the output 
resonant circuit tuned to the carrier frequency must be much wider than the modulating frequency 
bandwidth to present purely i? L for complete AM signal. Thus, the useful information within the 
complete AM signal, which is represented by P sb , defined as a ratio to the maximum possible 
transmitting power P max , is a function of the modulation factor m according to 

P, w, 2 

(6.9) 



P max 2(1 +mf 

being equal to P sb = 0.1 25P max for modulation factor m — 1 . This means that the total sideband power 
containing all of the information does not exceed 12.5% in a maximum total AM-signal power that 
can be potentially transmitted. According to Eq. (6.8), each of two sideband components contains one- 
fourth as much power as does the carrier, so that the power required to transmit the carrier-frequency 
signal is at least twice greater than the useful power corresponding to the sidebands. Moreover, for 
complicated AM signals, the long-time average modulating factor is usually significantly smaller 
than its maximum short-time value, thus resulting in a significant reduction of the sideband power in 
an overall transmitting AM-signal power. Therefore, despite the simplicity of the AM approach, the 



AMPLITUDE MODULATION 259 



overall AM transmitter efficiency is sufficiently small, especially when the linear signal transmission 
with negligible nonlinear distortion is required. 

An amplitude-modulated wave possesses distortion to the extent that the modulation envelope fails 
to reproduce exactly the modulating signal. This distortion can be classified as amplitude (nonlinear), 
frequency, or phase (time-delay) distortion [3]. Amplitude distortion exists when the modulation 
envelope contains frequency components not present in the modulating signal due to the inherent 
nonlinearity of the active device being modulated. This results in harmonics of the modulating signal, 
which in turn denotes the presence of higher order sideband components in the output spectrum. 
Frequency distortion arises when the degree of modulation produced by a modulating signal of given 
amplitude varies with the frequency of the modulating signal, when the sideband components of 
different frequencies do not have the correct relative amplitudes. Time-delay distortion occurs when 
the phase relationships between different frequency components of the modulation envelope differ 
from the phase relationships that exist in the modulating signal. 

Distortion in an AM signal can arise either from imperfections in the modulation system that 
produce this signal, or from the action of the active and passive circuits that transmit AM signal. 
Thus, when an AM signal is applied to a tuned circuit resonant at the carrier frequency, the upper and 
lower sideband frequencies can be reduced in amplitude symmetrically by an amount that increases the 
higher the modulating frequency. At the same time, the sideband frequencies undergo symmetrical 
phase shifts that introduce a time delay. If the carrier frequency does not coincide with resonant 
frequency of the tuned circuit, then the upper and lower sidebands are characterized by unequal 
transmission because of the asymmetrical phase shifts with respect to the carrier. This introduces 
quite severe amplitude distortion of the modulation envelope. 



6.1.2 Amplitude Modulators 

Generally, an AM can be generated by multiplication of the carrier and modulating signals in a 
nonlinear circuit. For example, if the transfer volt-ampere characteristic of a nonlinear device (diode 
or transistor) is represented by a second order polynomial 

i(v) = a 0 + a t v + a 2 v 2 (6.10) 

then, by neglecting the effect of the output voltage on the output current and substituting the sum of 
the carrier and modulating signals 

v = v 0 + v m = V 0 cosa> 0 t + V m cos Qt (6. 1 1) 

into Eq. (6.10), the output current i(v) can be written as 

i(v) — flo + ai (Vb cos (Mo? + Vm cos Q.t) + «2 ( V 0 2 cos 2 mot + 2V m Vb cos Qt cos a>ot + v£ c °s 2 ^f) • 

(6.12) 

The output current given by Eq. (6.12) can be rewritten as a sum of the harmonic components 
with different frequencies using trigonometric identities by 

i(v) = a 0 + y (V 0 2 + V m ) + a x (Vocoscnof + V m cos£2f) + a 2 V m V 0 [cos(cn 0 + £2) + cos(&> 0 - &)] 

+ y (V 0 2 cos2£,;of + V, 2 cos2^) (6.13) 

whose spectrum, containing the carrier frequency co 0 and its second harmonic component 2cu 0 , 
the modulating frequency Q and its second harmonic component 2Q, and the sum and difference 
intermodulation products ojq ± ^, is shown in Figure 6.4(a) [4]. 



260 MODULATION AND MODULATORS 





H\\ 





















o d 2n <o„-n (o 0 w„ + n 

(a) 

/' 4 IU\ 



0 fi 2fl3fi W0-2Q a, e)o + 2CI 2<o 0 - Q 2co 0 2w 0 + Q 3w 0 M 

FIGURE 6.4 Output AM current spectra. 

In order to obtain the required AM signal, it is necessary to represent only the frequency compo- 
nents co 0 , co 0 + Q, and a) 0 — Q from the whole spectrum. This can be achieved by using a resonant 
circuit tuned to the carrier frequency co 0 with a certain frequency bandwidth (BW) when the portion 
of the output current flowing to the load can be obtained from Eq. (6. 12) as 

i' L = a\ Vb cos wot + 2fl2 V m Vo cos Qt cos eotf . (6. 14) 

As a result, if the bandwidth of the resonant circuit is such that the load is seen as pure resistance 
i?L for frequencies co 0 , a> 0 + Q, and co 0 — Q and is zero for the rest frequency components, then the 
AM voltage across the resonant circuit is defined by 

v™ 1 = ai R L V 0 ^1 + ^-^cosfifj cosco 0 t (6.15) 

which can be simply rewritten as 

v£ M = V L (1 +m cos tit) cos co 0 t (6.16) 

where K L = ai^ L V 0 is the carrier amplitude and m — 2(a 2 /ai)V m is the modulation factor. In this 
case, larger values of the modulation factor m correspond to stronger device nonlinearity defined by 
ct2 and greater modulating amplitude V m . Since the modulation factor in is directly proportional to 
the modulating-signal amplitude V m , then ideally the modulation process is distortion free. 

However, if the transfer volt-ampere characteristic of a nonlinear device is represented by a 
third-order polynomial i(v) — a 0 + aiV + a 2 v 2 + a 3 v 3 , then the output current spectrum will 
contain the three harmonic components of the carrier and modulating frequency, a>o and Q, and the 
intermodulation components of the second and third orders, a> 0 ± £2, co 0 ± 2£2, and 2(o 0 ± Q, as 
shown in Figure 6.4(b). To achieve a distortion-free AM signal, it is necessary to be confined only to 
the frequency components coq and a>o ± Q. However, in this case the bandwidth of the resonant circuit 



AMPLITUDE MODULATION 261 



is not narrow enough to suppress the second order intermodulation products a> 0 ± 2Q, resulting in a 
parasitic AM of the carrier frequency by the second harmonic of the modulating frequency 2Q. 

An AM can be easily realized by variation of the base- or gate-bias voltage of the transistor, which 
produces a corresponding variation in the amplitude of the transistor output voltage in accordance 
with the modulating voltage. However, despite the simplicity of implementation, some distortion (5% 
to 10%) of the envelope is inherent with this method because of the nonlinearities of the device input 
capacitance and transconductance as nonlinear functions of a bias voltage [5]. In addition, it is diffi- 
cult to provide a sufficient isolation between the carrier and modulating signal paths, especially if the 
spectrum of the modulating frequencies is wide enough, which is the case for modern telecommunica- 
tion modulation formats. To improve the linearity performance of the AM transmitting signal without 
additional linearization envelope feedback loops, it is best to apply an AM at low power levels using 
either cascode or differential-pair transistor stage, followed then by a linear power AM amplifier. 

Figure 6.5(a) shows the circuit schematic of an amplitude modulator based on a cascode where the 
bottom transistor in a common emitter configuration is used for carrier signal and the top transistor in a 




Modulation 
envelope 



Modulating 
signal 



FIGURE 6.5 Collector modulation using cascode. 



262 MODULATION AND MODULATORS 




''trinsl 

FIGURE 6.6 Collector modulation using differential pair. 



common base configuration is used for modulating signal. Such a cascode amplifier can operate as an 
amplitude modulator with linear modulation range of about 15-20 dB, because of the collector-bias 
voltage variation of the bottom transistor when the cascode amplitude characteristic V 0 ut(^mod) is 
linear over a significant range of the modulating amplitudes applied to the base of the top transistor, 
as shown in Figure 6.5(6). The significant isolation of the modulating and carrier paths can be easily 
achieved by using a proper RC low-pass filter connected to the base of the top transistor. As an 
alternative, the carrier frequency path can be connected to the top transistor, while the modulating 
signal is applied to the bottom device, providing a linear emitter modulation of the carrier signal. 
However, due to a sufficiently high collector-base device capacitance, the significant feedthrough 
carrier power flows to the load in this case. Similar or even better linearity of the output AM signal 
can be achieved using a differential-pair transistor stage where the carrier signal is applied to the 
current-source device, as shown in Figure 6.6. In this case, the modulating signal can be applied to 
one transistor of the differential pair, while the load for the resulting AM signal can be connected to 
the collector of the other differential-pair transistor. The modulating signal can also be applied to both 
differential-pair devices to obtain the differential output AM signals at the outputs of the transistors 
relatively to each other. The voltage range of the modulating amplitudes is set by a series resistor in 
the modulating signal path. 

6.2 SINGLE-SIDEBAND MODULATION 

The single-sideband (SSB) method of signal transmission was proposed by John Carson in 1915 as 
a result of pure mathematical studies related to modulation of a continuous-wave carrier by means 
of vacuum tubes. He was the first to recognize that the suppression of the carrier either without or 
with suppression of one sideband would uniquely define the transmitting message that eventually led 
to the development of the double-sideband amplitude modulation (DSB-AM) and single-sideband 
amplitude modulation (SSB-AM), respectively [6,7], 

6.2.1 Double-Sideband Modulation 

The usual method of suppressing the carrier component of an amplitude-modulated signal is to employ 
a balanced modulator, which can provide carrier compensation when combining two AM waves in 



SINGLE-SIDEBAND MODULATION 263 




FIGURE 6.7 Schematic of diode balanced modulator. 



a common load. In this case, the carrier voltage is applied with the same phase to the inputs of two 
single amplitude modulators, while the modulating signal is applied in opposite phases, usually by 
means of the center-tapped transformer. If the transfer volt-ampere characteristic of the nonlinear 
device is defined by Eq. (6.10), then the amplitude-modulated voltage in the load v^ M is defined by 
Eq. (6.16) for v — v 0 + v m . However, for v — v 0 — v m , it can be rewritten as 

= V L (1 - m cos Qt) cos m 0 t. (6.17) 

Thus, subtracting the load voltage given in Eqs. (6. 16) and (6.17) due to the back-to-back connected 
nonlinear devices, as shown in Figure 6.7 for a diode balanced modulator, results in a double-sideband 
output signal in the form 

^dsb _ q _|_ m CQS CQS 0) ^ t _ y L (\ _ m cos £2f)cosa>of = 2m Vl cos Q.t cos a>ot 

= raV L [cos(cu 0 + + cos(ft> 0 - Q)t] (6.18) 

whose resulting envelope varying at twice the modulation frequency with amplitude l2ml/ L cosf2fl 
is shown in Figure 6.8(a). In this case, the modulated waveform consists of two side-frequency 
components with no carrier. With complicated modulating signal, there will be upper and lower 
sidebands. Therefore, it is called the double-sideband or suppressed-carrier wave. 

The vector representation of this wave is shown in Figure 6.8(£>), with zero carrier vector and 
the side-frequency vectors rotating with angular velocities mo + f2 and a>o — Q, respectively. Here, 
similarly to AM signal, the side-frequency vectors are rotating in opposite directions with the angular 
velocity Q, and, in the complex plane, their phasor sum will always lie along the real horizontal axis, 
indicating amplitude variation but no frequency deviation. Since there is no carrier, all the transmitted 
power fully corresponds to the information transmission, whereas in ordinary AM less than one-third 
of the total power carries the modulating information. Both systems require a bandwidth equal to 
twice the highest modulation frequency £2 max . However, because the envelope of the DSB-AM wave 
is a full-wave-rectified replica of the modulating signal, the simple envelope detector cannot be used 
without first reintroducing the carrier. 

For a balanced modulator shown in Figure 6.7 with identical diode volt-ampere characteristics 
approximated by the third-order polynomial where the voltages across the top and bottom diodes 
are vq + v m and vq — v m , respectively (for simplicity, the voltage drops across the resistors are 
neglected), the corresponding diode currents !j and i 2 can be written as 

h Oo + «m) = "o + ai (v 0 + v m ) + a 2 (v 0 + v m ) 2 + a 3 (v 0 + v m f (6.19) 
( 2 (v 0 - v m ) = a 0 + a y (v 0 - v m ) + a 2 (v 0 - v m f + a 3 (v 0 - v m f (6.20) 



264 MODULATION AND MODULATORS 





BW 



0 Q 



(Dc. Q «)(, - Q 



2(l> : , - Q 2<A; + Q 



<<-> 



FIGURE 6.8 Time-domain, frequency-domain, and vector representation of DSB-AM wave. 



resulting in an output voltage 

u° SB = R L [(, (v 0 + v m ) - h (vo - v m )] = 2R L (aiD m + 2a 2 v 0 v m + 3a 3 vlv m + a 3 v^) . (6.21) 

From Eq. (6.21) it follows that the output voltage spectrum obtained by using trigonometric 
identities will contain much less frequency components than that of a single-diode modulator. For 
example, the dc, second modulating (2Q) and third carrier (3coq) harmonic components will be 
suppressed, as well as the intermodulation components of the second order co 0 ± 2Q, as shown in 
Figure 6.8(c), compared to the output AM spectrum shown in Figure 6.4(b). In this case, the double- 
sideband signal can be easily filtered by the output resonant circuits connected in parallel to each 
load resistor R^. 

Moreover, the further and significantly better harmonic suppression can be achieved by adding two 
additional diodes, thus resulting in a double-balanced diode modulator shown in Figure 6.9. In this 
case, the currents ;'i(i>o + v m ) and «2(i>o — Dm) are defined by Eqs. (6.19) and (6.20), while the currents 
*3( — i>o — "m) and U{— v o + v m) differ from the currents ;'i and i 2 by opposite signs of the voltages v 0 
and v m , respectively. Hence, the resulting output voltage at the output of a double-balanced modulator 
by adding the two output voltages, one caused by current ii and i% defined by Eq. (6.21) and another 
caused by currents i 3 and ; 4 defined by Eq. (6.21) with opposite signs for voltages v 0 and v m , can be 
written as 



= ('1 - k) + #l (h ~ U) = 2R L («it> m + 2a 2 v 0 v m + 3a 3 v%v m + a 3 v^) 
+ 2R L (-aiv m + 2a 2 v 0 v m - 3a 3 VgV m - a^) — ?,R^a 2 v 0 v m 



(6.22) 



SINGLE-SIDEBAND MODULATION 265 




FIGURE 6.9 Schematic of diode double-balanced modulator. 



which demonstrates the behavior of a double-balanced modulator as an ideal mixer-multiplier of two 
input signals with output spectrum containing the side-frequency components coq + Q and a>o — Q 
only. 

6.2.2 Single-Sideband Generation 

An SSB signal can be obtained by passing the output of a carrier-suppression system through the 
sideband filter that is sufficiently selective to transmit one sideband while significantly suppressing 
the other. For example, if the lowest modulating frequency for speech transmission is typically 
300 Hz, the filter characteristic must change from full transmission in its passband to very effective 
rejection in an interval of 2 x 300 Hz = 600 Hz. Even with well-defined filters such sharpness of 
the filter cutoff characteristic is possible only if the carrier frequency is low enough compared to 
the modulating frequency. Moreover, as the sharpness of cutoff is increased, the filter will introduce 
more phase and amplitude distortion in the portion of the transmitted signal near the cutoff region. 
Therefore, multiple-pole crystal or ceramic filters with adequately sharp cutoff characteristic can be 
used and frequency upconversion action can be applied if necessary for an SSB transmission at high 
radio frequencies [5]. 

The need for sharp cutoff filters can be avoided using a second method generally called the 
phasing method, which provides elimination of the unwanted sideband by phase cancelation. Its 
block diagram consists of the two balanced modulators and 90° phase shifters, which are necessary 
to deliver the input carrier and modulating signals for lower modulator with a phase difference of 
90° each compared to the upper modulator, as shown in Figure 6.10(a) [8,9]. As a result, if the 
phase requirements are satisfied exactly and the balanced modulators are identical, the combined 
outputs cancel for one sideband and add for the other. As a practical matter, it is quite easy to realize 
20-dB suppression, reasonable to expect 30 dB, and quite difficult to go beyond 40 dB. The main 
problem of this system is to provide an exact phase shift of 90° over entire bandwidth of modulating 
frequencies. To approximate the desired result physically, the modulating signal is passed through 
two networks having phase shifts that differ by 90° over the frequency range of interest while the 
attenuation difference is substantially constant [10]. 

The third method does not need either sharp cutoff filters or wideband 90° phase shifters. Fig- 
ure 6. 10(b) shows the circuit diagram of such an SSB system with the corresponding basic frequency 
relationships [1 1]. In addition, imperfections in the phasing or balancing do not result in the presence 
of the unwanted sideband in its usual location. Instead, the unwanted sideband occupies the same 
frequency band as the desired sideband, except that it is inverted. As it follows from Figure 6. 10(fo), 
the modulating signal flows to balanced modulators along with the corresponding quadrature com- 
ponents of a carrier frequency co 0 followed by the low-pass filters that remove the upper sidebands, 
respectively. The filter output components are combined in the other pair of balanced modulators 
with corresponding quadrature signals from a source of frequency co c , the band center of the final 



266 MODULATION AND MODULATORS 



cosf!) 



90° phase 



sin£2/ 




COSill COS(0,.f 



Com bin* r 



COS((0-j - 



SSB-AM 
output 



sinl>/ siiuon/ 



ViK{tfLj + 12)/ 



0081^+ Oir, - £!>! 

- tositu, - til;: - 12)/ 



4Jos((n e - ak i + £2)/ 




cos(o,\. - o», ■ £2)/ 
cos(m e ■ mi, nv 



FIGURE 6.10 Methods of single-sideband modulation. 



SSB frequency. As a result, the combined outputs from these balanced modulators cancel one pair 
of sideband components and add the other pair of sideband components to produce the desired SSB 
signal. As the entire circuit is bilateral, it can be used in demodulation as well as in generation of the 
SSB signals. 



6.2.3 Single-Sideband Modulator 

SSB modulators have numerous applications in a variety of existing and new communication systems 
including optical and millimeter waves. Basically, a practical SSB modulator represents an upcon- 
verter that generates an SSB suppressed-carrier (SSB/SC) output signal without the use of filters. 
Figure 6. 1 1(a) shows the block diagram of a typical high-frequency SSB modulator where either the 
lower sideband or the upper sideband can be selected by exchanging the in-phase (/) and quadrature 
(Q) modulating inputs [12]. Generally, the system should include two balanced modulators, an HQ 
generator to generate output quadrature modulating signals, a 90° branch-line hybrid combiner to 
input the carrier signals, and an in-phase Wilkinson combiner for summing the two output modulator 
signals. As an alternative, the upper sideband can be achieved by using an out-of-phase rat-race 180° 
subtracter at the output instead of a Wilkinson combiner. In microwave monolithic implementation, 
both the transmission-line branch-line coupler and Wilkinson combiner can be replaced by their 
lumped low-pass and high-pass filter equivalents based on monolithic spiral inductors and MIM 
capacitors [13]. To further minimize size and phase imbalance between modulator paths, the GaAs 
MESFETs or HEMTs as nonlinear elements rather than diodes can be used. 



FREQUENCY MODULATION 267 



0" Rl- 
input 



P. O- 



90° brandi- 
liiK coupler 



smcv 



Balanced 
modulator 



sinf!/ slntv 



i.r.ii' 



Mixinlating 
frequency 
]-'Q generator 




Ualancerl 
mud □ latin" 



ccisQr cos in,,? 



(a) 

SSB- AM 

OLlLpUl 




Modulaliny, inpuLs 
O f — 



rl 



SSU-AM 

oil Lp 111 



-O Ft 



input 



FIGURE 6.11 Microwave double-balanced single-sideband modulators. 



Figure 6. 1 1(b) shows the microwave monolithic GaAs MESFET SSB modulator circuit, operating 
with a 7-GHz carrier and providing better than a 17-dB suppression of any spectral products including 
carrier, undesired sideband, and third harmonic [12]. The circuit consists of two differential MESFET 
pairs, with the common source node of each pair biased through additional MESFET that acts as a 
constant current source. The drain node of each pair is connected together to add output signals from 
both differential pairs in phase. The quadrature carrier signals applied to the gate of one MESFET 
in each pair can be provided by a Lange coupler. The modulating audio signal is split into the / and 
Q components and applied to the gate of the second MESFET in each pair. The constant current 
source devices are necessary to balance the drain currents in both differential pairs, thus creating a 
double-sideband spectrum at the MESFET drains of each pair, connected together in parallel. Then 
by connecting in parallel both differential pairs, the SSB spectrum with upper sideband is realized. If 
the / and Q inputs are reversed in phase, the lower sidebands combine and the upper sidebands cancel. 



6.3 FREQUENCY MODULATION 



The subject of frequency modulation (FM) is well known a long time ago, though the early experiments 
in radio transmission appeared interested only in AM. In 1902, Fessenden proposed to use a condenser 



268 MODULATION AND MODULATORS 



type of microphone in circuits that appear to produce FM, but in his publication remarks about tests 
he seemed interested in using the frequency variation to produce amplitude variation by throwing the 
frequency in and out of resonance with the antenna or other tuned circuit to modulate the amplitude 
[14]. Moreover, in 1922, Carson published his famous paper pointing out that FM would not provide 
a narrower band than AM, and that it inherently produces distortion in the signal, and hence, the FM 
system is inferior to the AM system [15]. And only one and half decades later, Armstrong theoretically 
explained and experimentally demonstrated that, if one widened the swing in FM much beyond the 
bandwidth used in AM and used an effective limiter in the receiver, noise lower than the carrier could 
be substantially eliminated, thus proving a significant superiority of the FM broadcasting system over 
its AM counterpart [16]. 

6.3.1 Basic Principle 

Generally, the modulated signal can be written in the form 

v(t) = V(f)cos0(f) (6.23) 

which is fully described by a time-varying amplitude V(t) and a time- varying phase angle (p(t). If in 
AM the carrier envelope V(t) is varied while 4>{t) remains constant, then in angle modulation V(t) — 
Vq is fixed and the modulating signal varies cf>(t). Angular modulation can be either FM or phase 
modulation (PM), depending upon the exact relationship between <p(t) and the modulating signal. 
The instantaneous angular velocity co(t) can be written in a common case as 

dd>(t) 

co(t) = lit f{t) = (6.24) 
at 

from which the instantaneous phase can be found by integrating according to 

t 

<j)(t) = J (jo(t)dt + (6.25) 
o 

where <po is the initial phase at t — 0. 

A frequency-modulated signal with cosine modulation when the instantaneous modulating signal 
varies in accordance with v m (t) — V m cos£2t represents by definition a signal in which the instantaneous 
angular velocity is varied according to 

Vm(t) 

m(t) = a>o H Am = coq + 2n Af cosQt (6.26) 

Vm 

where cdq is the carrier or average radian frequency, £2 = 2jr/ m is the modulating radian frequency, 
and Af is the maximum deviation of instantaneous frequency from average obtained at time moments 
when v m — V m . A fundamental characteristic of a frequency-modulated signal is that the maximum 
frequency deviation Af is proportional to the peak amplitude of the modulating signal V m and is 
independent of the modulating frequency. Integrating Eq. (6.26) gives 

Af 

0(0 = a>ot H -smQt + (p 0 (6.27) 

fm 

where for simplicity it can be assumed that <p 0 — 0. 

Consequently, substituting Eq. (6.27) in Eq. (6.23) results in 

v(t) = V 0 cos (w 0 t + — sinfif ) (6.28) 



FREQUENCY MODULATION 269 



which is commonly written for FM as 

v(t) = VocosOwo/ + msinSlt) (6.29) 

where 

m = k — = — (6.30) 

,/m Jm 

is called the modulation index for frequency modulation, where k is the factor of proportionality 
[4]. Note that, for a given frequency deviation A/, the modulation index m varies inversely as 
the modulating frequency f m . Time-domain representation of the FM process with a sinusoidal 
modulating wave is shown in Figure 6.12. As the modulating signal swings positive, the carrier 
frequency is increased, reaching its highest frequency at the positive peak of the modulating signal. 
When the signal swings in the negative direction, the carrier frequency is lowered, reaching a minimum 
when the signal passes through its peak negative value. 

Equation (6.29) can be rewritten through trigonometric expansion as 

v(t) — Vb cos (msinfif) cos co 0 t — V 0 sin (msinQt) sin a) 0 t (6.31) 

which represents a sum of two quadrature signals with a carrier frequency coq, each of which is 
amplitude-modulated with a modulating frequency f2. Generally, two different classes of FM signals 
can be distinguished, depending on the values of the modulation index m [17]. For m < 0.5, it is 




,/M 



FIGURE 6.12 Frequency modulation process in time domain. 



270 MODULATION AND MODULATORS 



called the narrowband frequency modulation, and Eq. (6.31) can be approximated as 



m Vo m Vo 
v(t) = Vq cosa>o? — m Vbsin£2f sinc^f = Vo cos co 0 t H ^~ cos (^0 + ^) f ^~ cos ( m o — fH)t 



(6.32) 



since cos(m sinfi/) == 1 and sin(ra sinfif) = m sinfif. Hence, the bandwidth of a narrowband FM is 
approximately 2Q, which is the same as for AM signal containing the carrier frequency coq and two 
sideband frequencies a> 0 + Q and co 0 — Q. In this case, the amplitudes of the sideband frequencies 
are defined by a modulation index m. However, the narrowband FM spectrum differs from the AM 
spectrum due to opposite sign in the second term, resulting in its phase shift by 180°. 

For m > 0.5, the frequency-modulated process is called the wideband frequency modulation, which 
is the type of FM most often used in analog communication systems. To determine its bandwidth 
requirements, it is necessary to expand the terms cos(m sinfif) and sin(m sin£2f) in Eq. (6.31) as the 
even and odd periodic functions in a Fourier series according to 



cos (msinQt) = J 0 (m) + 2J 2 (m) cos (2Qt) + 27 4 (m) cos (4Qt) H (6.33) 

sin (msinQt) = 2J x {m) sin (£2t) + 2J 3 (m) sin (3Qt) H (6.34) 



where J n (m) are the Bessel functions of the first kind. 

Substituting Eqs. (6.33) and (6.34) into Eq. (6.31) and using trigonometric identities result in 



v(t) — VqIJq (m) cos co 0 t + J i (m) [cos (a> 0 + Q) t — cos (a> 0 — f2) t] 
+ Ji (m) [cos (co 0 + 2£2) t + cos (co 0 — 2Q) t] 
+ ^3 (m) [cos (w 0 + 3£2) t — cos (a> 0 — 3ST2) /] + ••■} 



= Vo 



7o (m) cos coot + J n (m)Ji (m) [cos (coq + n£2) t + (—1)° cos (a>o — n£i) /] > (6.35) 



from which it follows that the FM signal spectrum is discrete, symmetrical relatively to the carrier 
frequency coo, and contains an infinite number of sideband frequencies spaced at radian frequencies 
± SI, ± 2Q, . . . , about the carrier with amplitudes V„ = V Q J K (m). The plots and numbers of the 
relative carrier and sideband-frequency amplitudes for different modulation indexes m are shown in 
Figure 6.13. It is clearly seen that, as the modulation index varies, a carrier frequency or a sideband- 
frequency pair may vanish entirely. This phenomenon can be used to set the frequency deviation of 
an FM transmitter, for example, by choosing the modulating frequency / m for a specified frequency 
deviation Af so that Jo(m) — 0. 

The characters of the frequency spectra obtained under different conditions with FM are shown 
in Figure 6.14 (fixed f m with increasing m) and Figure 6.15 (fixed Af with decreasing/ m ). Although 
the bandwidth occupied by the frequency-modulated signal is theoretically infinite, in reality the 
amplitudes of higher order sideband frequencies decrease rapidly. When the modulation index is less 
than 0.5, the second-order sideband frequencies are relatively small, and the frequency bandwidth 
(BW) required to accommodate the essential part of the signal is the same as in AM when BW = 2/ m . 
Besides, the amplitude of the first-order sideband frequencies is almost exactly proportional to the 
modulation index. On the other hand, when the modulation index exceeds unity, there are important 
higher order sideband components contained in the FM signal, and, for very large m, the approximate 



FREQUENCY MODULATION 271 

Relative amplitude 

1.0 

08 
06 
04 
02 
0 
02 
o 4 

123456789 

Modulation index 



Modulation 
index 


Camet 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


0 


1 00 






























0 25 


0 98 


0 12 




























0 5 


094 


0 24 


0 03 


























1 0 


0 77 


044 


0 11 


0 02 
























1 5 


0 51 


0 56 


0 23 


0 06 


001 






















0 


0 22 


0 58 


0 35 


0 -3 


0 03 






















2 41 


0 


0 52 


0 4? 


0 20 


0 06 


0 02 




















2 5 


0 05 


0 50 


0 45 


0 22 


0 07 


0 02 


0 01 


















3 0 


■0 26 


0 34 


0 49 


0 31 


0 13 


004 


0 01 


















i 0 


0 40 


■0 07 


0 36 


: 43 


0 28 


0 13 


0 05 


D 02 
















5 0 


-0 18 


-0 33 


0 05 


0 36 


0 39 


0 26 


0 13 


0 05 


0 02 














553 


0 


•0 34 


•0 13 


0 25 


0 40 


0 32 


0 19 


0 09 


0 03 


0 01 












6 0 


0 15 


0 28 


-0 24 


: 


0 36 


0 36 


0 25 


0 13 


0 06 


0 02 












70 


0 30 


0 


■0 3 


-0 1 7 


0 16 


0 35 


: 34 


0 23 


0 13 


0 06 


0 C2 










80 


0 17 


023 


-0 11 


•0 29 


-0 10 


0 19 


a 34 


032 


0 22 


0 13 


0 06 


0 03 








E 65 


0 


0 27 


006 


■0 24 


-0 23 


0 03 


0 26 


034 


0 28 


0 18 


0 10 


0 05 


0 02 






90 


-0 09 


0 25 


0 14 


-3 18 


-0 27 


-0 06 


0 20 


0 33 


0 3' 


0 21 


0 12 


0 06 


0 03 


0 01 




l( 


-0 25 


-■ 


0 25 




22 


23 


•0 01 


0 22 


:-. 


0 29 


: : ■ 


: •: 


0 06 




0 01 



FIGURE 6.13 Carrier and sideband amplitudes for FM signals. 



bandwidth requirements can be defined as BW = 2mf m — 2 A/. Figure 6.16 illustrates the analyzer 
display of an FM signal with m — 95 when it is required to use the entire spectrum width. 

Thus, a general rule of thumb for the approximate bandwidth of a frequency-modulated signal for 
any m is 

BW = 2(A/ + / m ) = 2/ m (m+l) (6.36) 

which is called the Carson's bandwidth rule. In other words, any frequency-modulated signal will 
have an infinite number of sidebands but in practice all significant sideband power (98% or more) is 
concentrated within the bandwidth defined by Eq. (6.36). However, setting the arbitrary definition of 
occupied bandwidth at 98% of the total power still means that the power outside the band is only by 
10 x logio(0.02/0.98) = 16.9 dB less than the in-band power. For voice communications a higher 




272 MODULATION AND MODULATORS 





1 fl a 

I At * 




in = 0.2 


0.5 ■ 
1 






1 


./0-./n, h /C+A ./ 



m - 1 



(I? 



m = 10 



I I I 



1 1 1 1 , 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 



X 



1 I I 



FIGURE 6.14 Frequency spectra of FM signal with fixed modulating frequency. 



.A % 



.A i A 

— H 



.'. -V, .' 



/«- 5 














1 1 1 l 




1 


1 


1 




1 I , ,. 



2 A/ 



-»- 10 



... Il II I ■ ll I I I I I I I I I . ill I 

,'ii-lt>',, f- fi-Wr> 1 

FIGURE 6.15 Frequency spectra of FM signal with fixed frequency deviation. 



FREQUENCY MODULATION 273 



RL:-31.3dBm 10 dBm/ AT 0 dB ST 15 s D:PK 





















— A 
I 




mi 










— L 
I 1 














— i — 

r 






1 

i — 
i 





CNF: 1999.999 4 MHz SPF:5 MHz RB1 KHz VB1 KHz 



FIGURE 6.16 Analyzer display of FM signal with m = 95. 



degree of distortion can be tolerated, and it is possible to ignore all sidebands with less than 10% of 
the carrier voltage. In broadcast FM, where Af is restricted to be 75 kHz and maximum / m should be 
less than or equal to 15 kHz, the smallest modulation index is m — 75 kHz/ 15 kHz = 5. As a result, 
from Eq. (6.36) it follows that BW = 180 kHz, which is safely within the total bandwidth of 200 kHz 
allocated to each FM station. 

More accurate approximation of the frequency-modulated signal bandwidth for m — 0 to 24 can 
be calculated for sinusoidal modulation as 

BW = 2/ m (m + Vm + l) (6.37) 

and for square-wave modulation as 



BW = 2/ m ^/ — -m + m 2 (6.38) 

with amplitudes less than 1% of the non-modulated carrier amplitude [18]. 

Generally, the relative carrier and sideband frequency amplitudes in a frequency-modulated signal 
will vary with amplitude and frequency of the modulating signal, but the total power contained in the 
modulated waveform remains constant. This is in contrast to AM, where the sideband amplitudes and 
the total power are controlled by the modulation, but the carrier amplitude is not. Besides, contrary to 
the situation in AM when each frequency component in the modulating signal produces a single pair 
of sideband frequencies, the superposition principle does not hold in a frequency-modulated signal 
for complex modulating signal, resulting in asymmetrical arrays of sideband frequencies, even with 
harmonically related modulating signals [4]. 

6.3.2 Frequency Modulators 

Generally, there are two basic methods to provide an FM process: direct method by varying the reactive 
element of the frequency oscillator and indirect method by proper phase varying in the phase modulator 
according to Eq. (6.27), which follows the fixed-frequency oscillator. Usually, varactor whose junction 
capacitance varies in accordance with applied reverse bias voltage is used as a frequency-tuning 
element in the oscillators. By providing a proper reverse bias voltage and low-frequency modulating 



274 MODULATION AND MODULATORS 





FIGURE 6.17 Schematic of varactor-tuned transformer-coupled oscillator. 



signal to the varactor that is a part of entire oscillator resonant circuit, the frequency-modulated signal 
at the output port of the oscillator is achieved. To evaluate the quality of such a frequency-modulated 
signal, it is very important to know the behavior of the varactor voltage-capacitance characteristic and 
its influence on the oscillator modulation characteristic, representing a dependence of the oscillation 
frequency on a slowly varying varactor bias voltage. Figure 6.17 shows the simplified transformer- 
coupled MOSFET oscillator circuit where the modulating signal v m — V m cos£2f is applied to the 
varactor included into the oscillator resonant circuit. 

The voltage-capacitance characteristic of the abrupt-junction varactor can be written as 



C v (V v ) : 



C 



yd 



1 + — 
<p 



(6.39) 



where V v is the fixed bias voltage, C v o is the varactor capacitance when V v — 0, and <p is the contact 
potential. Substituting the voltage increment v — V v + Ad instead of V v into Eq. (6.39) gives 



AC V 

~c7 



1 



(6.40) 



where £ = Av/(<p + V v ). Equation (6.40) can be expanded into a Taylor series as 



x — h ■ 

2 8 16 



(6.41) 



where x — AC V /C V . By assuming that the normalized voltage increment § represents the value of the 
modulating voltage as f = Av/(<p + V v ) — K m cos£2/, then Eq. (6.41) can be rewritten as 



AC V 

~c7 



3V 2 V m 3V 2 



16 



16 



(6.42) 



For a single resonant circuit oscillator with the radian frequency deviation Aco around the oscil- 
lation frequency w Q when varactor represents the only circuit capacitance, 



Aco ■■ 



VL (C v + AC V ) 



w 0 



1 + 



AC V 



-1/2 



(6.43) 



FREQUENCY MODULATION 275 



where coq — 1 /v LC y . As a result, from Eq. (6.43) it follows that 

Af 1 x 3x 2 

— = , -1 = + . (6.44) 

f 0 2 8 

Substituting Eq. (6.41) with two first factors only into Eq. (6.44) results in 

Af I 3f 2 

— = -- — + ■■• (6.45) 
fo 4 32 

which represents a nonlinear dependence of the frequency deviation Af versus voltage variation Au 
on the varactor. Then, substituting £ = V m cos£2f into Eq. (6.45) gives 

Af 3 V 2 V m 3V 2 

=±- = - — = + —cosQt - -^cos2£2r + • • ■ (6.46) 
fo 64 4 64 



with the frequency deviation 



A/=^/o (6.47) 



the second-harmonic nonlinear coefficient 



16 4 f 0 
and the relative shift of the average frequency 

A/avr = = 3 / Af 

fo 64 4 ^ f 0 



fc 2 =^ = ?¥ (6-48) 



(6.49) 



Generally, the function jc(§) in Eq. (6.41) depends on the voltage-capacitance characteristic of 
the particular varactor, whereas the function Af(x) is characterized by the particular type of the 
varactor connection to the oscillator resonant circuit. Figure 6.18(a) shows the general case of the 
partial varactor connection to the oscillator resonant circuit. When C\ 3> C2 + C v , it can represent a 
parallel resonant circuit with variable capacitance C v , as shown in Figure 6.18(6). When C2 <H C v , 
the oscillator can be represented by a series resonant circuit with variable capacitance C v , as shown 
in Figure 6.18(c). 



c, c, 




(a) (b) (c) 



FIGURE 6.18 Varactor connection to resonant circuit. 



276 MODULATION AND MODULATORS 



For a parallel resonant circuit with Co = C% + C v and varactor capacitance variation AC V due to 
At), 

AC C 2 + C V + AC V C v AC V 

y = = 1 = = px (6.50) 

Co C2 "I - C v C2 "I - C v C v 

where the total capacitance variation AC corresponds to the varactor voltage variation Ad and 
p — C V /(C2 + C v ) is the coefficient of partial varactor connection into the parallel oscillator resonant 
circuit. 

For a series resonant circuit with Co = C v Ci/(C v + Ci), 

= AC = (1+xHd + Cy) px -<\- W 2 + (6 51) 

y Co d + C v (l+x) \ + {\-p)x PX 1 P 

where p — Ci/(Ci + C v ) is the coefficient of partial varactor connection into the series oscillator 
resonant circuit. 

The general-type resonant circuit shown in Figure 6.18(a) represents a combination of the series 
and parallel resonant circuits. The relative deviation yi of the capacitance C' v — C% + C v can be 
defined from Eq. (6.50) as 

ac; 

xi = -— = p\x (6.52) 

where p\ — C V /(C2 + C v ). In this case, the capacitance C v is the variable capacitance for the circuit 
including the capacitance C\ and inductance L. Therefore, using Eq. (6.51), 

y(x) = p 2 xi - p 2 (l - Pi)x\ H (6.53) 

where p 2 = Ci/(C] + C;) = C^C, + C 2 + C v ). Substituting Eq. (6.52) into Eq. (6.53) and 
representing Eq. (6.53) as y(x) = a\x + a 2 x 2 + ■ • • allow the first two coefficients to be determined 
asai = pip 2 and a 2 = —p\p2 (1 - P2X 

As a result, by choosing the coefficients p\ and p 2 with proper values of C\, C%, and, C v , the 
behavior of the oscillator modulation characteristic can be optimized for linear frequency tuning for 
the particular varactor voltage-capacitance characteristic [19]. 

Generally, the voltage v v across the varactor shown in Figure 6.19 consists of the dc bias voltage V v , 
low-frequency modulating component v m with amplitude V m , and high-frequency carrier component 
i>o with amplitude Vo, which can be written as 

v v - V v + V m cos fir + Vb cos co Q t (6.54) 

where fi is the modulating frequency and coq is the carrier frequency. The bias voltage V v is usually 
chosen at the center point of the varactor voltage-capacitance characteristic to maximize the frequency 
tuning bandwidth without performance degradation due to the forward conduction of the active device 
or its breakdown. Under these assumptions, the borders of voltage variations across the varactor 
junction should satisfy the condition V m + Vo — 0.5Vvmax- 

Figure 6.20 shows the schematic of the linearized direct frequency modulator based on a crystal 
transistor oscillator with audio-frequency input port, which is intended to use in the land mobile 
radio of low VHF band [20]. Assuming that the crystal fundamental frequency may be selected on a 
cost-size basis only, a multiplier ratio of 3 will require a crystal fundamental frequency of 10 to 17 
MHz to cover the 30 to 50 MHz VHF band. To provide a 5 kHz deviation with a multiplication ratio 
of 3, the varactor-tuned crystal oscillator must be capable of a ± 1 .7 kHz deviation at the fundamental. 
In this case, the bias voltage must be large enough to insure the varactor diode remains reverse biased. 



FREQUENCY MODULATION 277 




6.20 Schematic of linearized direct frequency modulator. 



278 MODULATION AND MODULATORS 



The tank circuit in the collector is tuned to the third harmonic of the fundamental crystal frequency, 
which is then amplified without further multiplication straight through antenna. The linearization 
of the modulation characteristic can be achieved by optimizing the value of the variable inductor 
connected in parallel to the varactor diode. The value of the shunt capacitor at the audio input is 
chosen to look like an RE short to the inductor but still have relatively high reactance over the 
audio band up to 3000 Hz. By proper choice of both the inductance and the varactor capacitance, 
the undesirable modulation nonlinearity can be considerably reduced. The linearizing network for a 
microwave frequency modulator based on the varactor-tuned cavity or dielectric resonator transistor 
oscillator can include the optimized combination of a shunt variable capacitor and the high-impedance 
and low-impedance transmission lines [21]. 



6.4 PHASE MODULATION 

In order to introduce the PM principle, first consider the carrier cycle as the projection of a point 
rotating counterclockwise in a circle where the phase at any given point is the angle between the start 
point and the point on the waveform, as shown in Figure 6.21. This is called the vector representation, 
which can also be plotted as the amplitude against the number of degrees of the point rotation. For 
each cycle of the earner, the point rotates in one complete circle, or period of 360°: starting from 0°, 
rotating through 180°, and coming back to 0°, where the next cycle begins. The position of the point 
at any instant can be indicated by a vector drawn from the center of the circle, showing the particular 
case of 45° in Figure 6.21. Note that this vector is rotating at the carrier frequency a>o. 

A phase-modulated signal with cosine modulation (instantaneous modulating signal varies in 
accordance with cosfif) will represent by definition a signal in which the instantaneous phase is 
varied according to 

v m (t) 

4>{t) — (Dot H A(p + (p 0 — a> 0 t + A(p cosfif + 0o (6.55) 

where a> 0 is the carrier or average radian frequency, f2 = 2nf m is the modulating radian frequency, <p 0 
is the initial phase, and A(p is the maximum deviation of instantaneous phase from its average value. 
Similarly to a frequency-modulated signal represented by Eq. (6.27), Eq. (6.55) for a phase-modulated 
signal can be rewritten as 

<p(t) — a> 0 t + m cosQt + <p 0 (6.56) 
where m — kV m — A(p is called the modulation index for phase modulation [4]. 




FIGURE 6.21 



Projection of circulating vector point to form sine-wave cycle. 



PHASE MODULATION 279 



Substituting Eq. (6.56) into Eq. (6.23) with constant carrier amplitude Vq results in a phase- 
modulated signal with cosine modulation written as 

v(t) — Vbcos(&>of + mcosQt) (6.57) 

where it is assumed that (p 0 — 0 for the sake of simplicity. A comparison of Eqs. (6.29) and (6.57) 
shows that the phase-modulated signal contains similar sideband components as does the frequency- 
modulated signal, and if the modulation indexes in these two cases are the same, the relative amplitudes 
of their corresponding components will also be the same. The frequency of the phase-modulated signal 
obtained from Eq. (6.56) varies according to 

d(p(t) 

ai — = co 0 — mQsinQt. (6.58) 

dt 

As a result, from a comparison of Eqs. (6.56) and (6.58) it follows that the cosine varying of the 
signal phase causes the sinusoidal varying of the signal frequency. In this case, the phase modulation 
with maximum phase deviation A<p is accompanied by the frequency modulation with maximum 
frequency deviation Af — f m A<t>. At the same time, as it follows from a comparison of Eqs. (6.26) 
and (6.27), the FM with maximum deviation Af is equivalent to the PM with maximum phase 
deviation A<f> as 

Af 

M>=~-. (6.59) 

fm 

Consequently, varying the modulating frequency f m affects differently the frequency spectra of 
FM and PM signals. For example, the modulation index of FM signal increases by reducing the 
modulating frequency, as shown in Figure 6.15; however, the spectrum bandwidth remains almost 
the same. On the contrary, the spectrum bandwidth of FM signal reduces with increasing modulating 
frequency, as shown in Figure 6.22 where the modulating frequency f m becomes three times smaller, 



w = 5 



Ll 



./n-5/.„ 



2M' 



h - V- f<> fi - s/„ f 
\ — w _^ 

FIGURE 6.22 Frequency spectra of PM signal with fixed modulation index. 



280 MODULATION AND MODULATORS 




(/>) 

FIGURE 6.23 Phase modulation process in time domain. 

but has no effect on the modulation index of PM signal. Time-domain representation of the PM 
process with a cosine modulating wave is shown in Figure 6.23. If the phase-modulated wave is 
shifted by 90°, it becomes looking alike the frequency-modulated wave shown in Figure 6.12. 

Thus, the common feature of both FM and PM is that the frequency varying inevitably results 
in a corresponding phase varying and vice versa, and, to produce a phase-modulated signal that is 
identical in phase with a frequency-modulated signal, the modulating voltages must differ in phase 
by 90°. However, the difference between FM and PM is that the maximum phase deviation for a 
harmonic FM is reversely proportional to frequency of the modulating signal according to Eq. (6.57), 
whereas the maximum frequency deviation for harmonic PM is directly proportional to frequency of 
the modulating signal. For constant amplitude of the modulating frequency, the maximum frequency 
deviation for a harmonic FM is constant, whereas it is linearly proportional to the modulating 
frequency for a harmonic PM, increasing with velocity of 6 dB per octave. 

The PM process can therefore be used to generate a true frequency-modulated signal by arranging 
so that the amplitude of the modulating voltage actually used to produce the phase variation is the 
signal modified by passing through the integrating resistance-capacitance network, in which the 
transmission is inversely proportional to frequency, instead of being the actual modulating signal. If 
the voltage appearing at the output of such a network is then used to control the instantaneous phase, 
the result is exactly the same as if the original signal were employed to control the instantaneous 
frequency [3]. Figure 6.24(a) shows the block schematic of an FM modulator where the frequency- 
modulated output signal is achieved by applying the modulating signal to the phase modulator 
input through the integrating network to reduce modulating amplitude versus increasing modulating 
frequency. Similarly, Figure 6.24(b) shows the block schematic of a PM modulator where the phase- 
modulated output signal is a result of applying the modulating signal through the differential network 
to increase the modulating amplitude versus increasing modulating frequency. 

An important practical consideration is the fact that the average power in an angle-modulated signal 
is constant. The average power in a band-limited signal is found by adding the power contributions 



PHASE MODULATION 281 



coscV 



Phase 
modulator 



FM 



___! 

Q 



slnQt 



K m cosQf 



(a) 



COSOV 



Frequency 
modulator 



V m cosLlt 



PM 



Qf-',,,sinQ; 



d 

7/T 



FIGURE 6.24 Indirect methods of frequency and phase modulations. 



of the various frequency components delivered to the load resistance R L as 




(6.60) 



by using Parseval's theorem according to which the average power is proportional to the sum of the 
squares of the individual Fourier components of the modulated wave. 

Figure 6.25(a) shows the circuit schematic of the phase modulator based on a three-section low- 
pass filter having identical resonant circuits, each tuned by the varactor. For a proper tuning, such 
a phase modulator can provide the phase deviation up to 30° for each section when the parasitic 
AM is negligible and nonlinear distortions are sufficiently small [22]. The phase modulator, whose 
basic reactive elements compose a second-order phase-shifting four-pole network, is shown in Figure 
6.25(6). Here, the low-frequency modulating voltage applied directly to the varactors changes their 
capacitances so that, for a proper dc biasing and sufficiently small capacitance variation, the amplitude 
variations are negligible with the phase deviation up to 70°. 

Figure 6.26 shows the simplified schematics of the reflection-type varactor-tuned phase modulator 
using either (a) quadrature branch-line hybrid or (b) 3-dB coupled-line directional coupler. In both 
cases, the input carrier signal divides equally between the two opposite ports of the hybrid or coupler, 
each of which is connected to the varactor. The varactor diodes are both biased in the same forward 
or reverse biased state, so that the resulting modulated waves reflected from these two reactive 
terminations will add in phase at the output port, which originally represents the isolated port. To 
isolate the high-frequency carrier and low-frequency modulating paths, the quarterwave transmission 



282 MODULATION AND MODULATORS 



Input 

O— 

On 



Vs/V^-AAA, — 1| 





zx 

Hh 



Output 

— o 

PM 



Input 
O 



(fl) 



Input 

o- 

('<<-, 



C 



Outpul 

— o 



PM 



Input 

— O 



n 



V^r-MM — of. 



FIGURE 6.25 Schematics of varactor-tuned phase modulators. 



lines can be used instead of lumped LC circuits. To improve the modulation linearity, a pair of 
varactors that are reversely biased with different voltages and separated by microstrip lines with 
optimized lengths to maximize reflection can be configured instead of a single varactor [23]. In 
this case, the phase deviation of 90° with linearity of 5% had been achieved within the frequency 
bandwidth of 0.8 to 1.2 GHz [24]. Much broader bandwidth can be achieved by using a 3-dB tandem 
or multisection Lange-type coupler. Three cascaded sections (with tapered varactor size), each of 
which is based on the lumped reflection-type quadrature hybrid and two varactors, could provide an 
X-band phase modulation with peak phase deviation of 300° and linearity of ±2° [25]. 



DIGITAL MODULATION 283 



Input VA 0"<P"< 




Input, fi 



(a) (b) 

FIGURE 6.26 Microstrip reflection-type phase modulators. 



6.5 DIGITAL MODULATION 

Any modulated signal described by Eq. (6.23) can be represented in polar form by 

v(t ) - V(t) cos [coot + <p(t)] (6.61) 

where V(t) is the time-varying amplitude (or envelope), coq is the carrier radian frequency, and 
<p(t) is the time-varying phase. By using the trigonometric identities, Eq. (6.61) can be rewritten in 
rectangular form by 

v(t) = /(f) cos co 0 t - Q(t) sin co 0 t (6.62) 

where I(t) — V(t)cos(p(t) is the in-phase time-varying signal component and Q(t) — V(t)sin(p(t) is 
the quadrature time-varying signal component, as shown in Figure 6.21(a). These components are 
orthogonal and do not interfere with each other. On a polar diagram, the /-axis lies on the zero degree 




FIGURE 6.27 Vector representation and block implementation of HQ modulator. 



284 MODULATION AND MODULATORS 



phase reference, and the g-axis is rotated by 90 degrees. The signal vector projection onto the /-axis 
is its in-phase (/) component, while the projection onto the g-axis is its quadrature (Q) component. 
In this case, the corresponding signal magnitude is defined as 

A = x// 2 + Q 2 (6.63) 

whereas the signal phase is obtained by 

^ = tan _1 — . (6.64) 

Since two baseband signals I(t) and Q(t) modulate two exactly 90° out-of-phase carriers cosft) 0 f and 
— sincoo/, respectively, then the system operating according to Eq. (6.62) is called the I/Q modulator. 
The operating principle of an HQ modulator, the block schematic of which is shown in Figure 6.27(b), 
is based on the splitting of the carrier signal into two equal signals when one flows directly to the 
balanced modulator to form the /-channel and the other flows into the other balanced modulator 
via 90° phase shifter to form the g-channel. The baseband /(f) and Q(t) signals (either analog or 
digital) modulate appropriately the carrier to produce the / (in-phase) and Q (quadrature) frequency 
components, which are finally combined to produce the desired radio transmitting signal. Since any 
RF signal can be represented in the HQ form, any modulation scheme can be implemented by an HQ 
modulator. Digital modulation is easy to accomplish with the HQ modulators because most digital 
modulation schemes map the data to a number of discrete points on the HQ plane, resulting in a 
simultaneous amplitude and phase modulation. 

To compare different modulation format efficiencies, it is necessary to first introduce the terms 
"bit" and "symbol," and then to understand the difference between bit rate and symbol rate. A bit 
is a binary digit as a basic unit of information capacity, taking a value of either 0 or 1. A symbol 
is a state or significant condition of the communication channel that persists for a fixed period of 
time. A sending device places symbols on the channel at a fixed and known symbol rate, which is 
measured in baud (Bd) or symbols per second. The baud unit is named in honor of Emile Baudot, 
the inventor of the Baudot code for telegraphy. If the transmitted pulses take on only two possible 
levels, each pulse represents 1 bit, and the bit rate = symbol rate = 1/r bits/s, where t is the width 
of rectangular pulse. However, when raised cosine pulses are used for data transmission, then the 
pulse rate or symbol rate = 1/r symbols/s. Generally, the basic difference is that the signal bandwidth 
needed for the communication channel depends on the symbol rate, and not on the bit rate. The bit 
rate is the frequency of a system bit stream. The symbol rate is the bit rate divided by the number of 
bits transmitted with each symbol. If one bit is transmitted per symbol, then the symbol rate would be 
the same as the bit rate. However, if two bits are transmitted per symbol, then the symbol rate would 
be half of the bit rate. If more bits can be sent with each symbol, then the same amount of data can 
be sent in a narrower spectral bandwidth. This is why modulation formats that are more complex and 
use higher number of states can send the same information over a narrower frequency bandwidth. 
For example, for 8-state phase-shift keying modulation, there are eight possible states that the signal 
can transmit at any time. Since 8 = 2 3 , there are three bits per symbol, which means that the symbol 
rate is one-third of the bit rate. 

6.5.1 Amplitude Shift Keying 

Amplitude shift keying (ASK) is a form of digital modulation that represents digital data as pulsed 
variations in the amplitude of a carrier wave. The amplitude of an analog sinusoidal carrier signal varies 
in accordance with the bit stream (modulating signal), keeping frequency and phase constant. In the 
modulated signal, logic 0 is represented by the absence of a carrier, while logic 1 is represented by some 
carrier amplitude, thus giving on-off keying operation. The ASK technique is also commonly used 



DIGITAL MODULATION 285 




Binary 
sequence 



OOK 
siynal 



0.5 



j L 



fo 



J_L 



1 



fo - 7/ ra f 0 - 5f m /o - 3/ m fo -f m fo +f m f 0 + 3/ m fo + 5f m fo + 7/m f 
FIGURE 6.28 Binary modulating signal, OOK signal and spectrum. 



to transmit digital data over optical fiber. For the fiber optic LED (light-emitting diode) transmitters, 
binary 1 is represented by a short pulse of light and binary 0 by the absence of light. On-off keying 
(OOK) is the simplest type of ASK modulation representing digital data as the presence (peak value) 
or absence (zero value) of a carrier wave. OOK is most commonly used to transmit Morse code over 
radio frequencies; the term "keying" is a historical remnant of telegraph transmission days. When 
more than two-level pulses are used, called the m-ary transmission for m-level signals, the modulation 
process is usually called the m-ary amplitude shift keying (MASK). 

Figure 6.28(a) shows the binary sequence 10010 that generates the OOK signal in the form of an 
RF pulse train. The OOK modulation can be considered as a modulation of the carrier by the periodic 
rectangular pulses with the symbol rate f m — l/(2r) = 0.5 Bd, where r is the symbol (bit) duration 
time. Figure 6.28(b) shows the OOK frequency spectrum where its spectral components are defined 
using a Fourier transform of the OOK signal with unit carrier amplitude as 



1 nir 

a 0 = 0.5, a„ = — sin — . 

nil 2 



n = 1, 3, 5, . 



286 MODULATION AND MODULATORS 



The OOK modulators can be used in modern digital communication systems where it is neces- 
sary to transmit an RF pulse-width train instead of the signal with non-constant envelope for highly 
efficient operation, and they play an especially important role in subscriber radio systems employ- 
ing time-division multiple access (TDMA) technology. The simplest implementation of the OOK 
modulators at microwave and millimeter-wave frequencies is based on using the different types of 
quarter- wavelength transmission lines and p-i-n diodes [26]. Figure 6.29(a) shows the typical circuit 
schematic of the OOK modulator based on a single-ended amplifying transistor stage that can be 
used at radio frequencies. Here, both the carrier signal and binary pulse sequence are delivered to 
the common transistor base, being isolated from each other by a low-pass second-order RLC circuit. 




FIGURE 6.29 Schematics of transistor OOK modulators. 



I 

0 

III 

3 



DIGITAL MODULATION 287 

I | Binary' 

sequence 




4-ary 
signal 



4ASK 
signal 



FIGURE 6.30 Binary sequence and 4ASK modulation. 



In this case, it is necessary to optimally choose its elements to minimize the overall circuit transient 
response caused by the finite pulse fall and rise times and periodic damped oscillation process. A very 
convenient practical implementation, especially for monolithic circuits, can be based on a cascode 
circuit shown in Figure 6.29(b) where the bottom transistor in a common emitter configuration is 
used for carrier signal and the top transistor in a common base configuration is used for binary pulse 
sequence. The significant isolation of the pulse-modulating and carrier paths can be easily achieved 
by using an optimized low-pass RC filter connected to the base of the top transistor. 

Figure 6.30 shows the binary sequence, 4-ary digital signal, and 4ASK signal with the amplitudes 
that are defined as V\ — Vo[2i — (in — 1)] for i — 0, 1,2, ... ,rn and m > 4. It is interesting to note that 
simply by observing the resulting 4ASK signal alone, it is difficult to directly distinguish this signal 
from the conventional binary ASK (BASK) signal with a modulation index m — 0.5. However, each 
RF pulse of the 4ASK signal, corresponding to a particular binary code, is characterized by different 
initial phase compared to the RF pulse with the same amplitude in a conventional binary case. 



6.5.2 Frequency Shift Keying 

Frequency shift keying (FSK) is an FM scheme in which digital information is transmitted through 
discrete frequency changes of a carrier wave. The simplest FSK is a binary frequency shift keying 



288 MODULATION AND MODULATORS 




Binary 
sequence 



BFSK 
signal 



0.5 . 








A ~ Vm 




J 






A + % 


> 4/m 1 
■ 1 










1 ,' /;, + 4/m 



A-3f m A-A; A+f m /o + 3/„ / 

(A) 

FIGURE 6.31 Binary modulating signal, BFSK signal, and spectrum. 



(BFSK). For a binary sequence, FSK simply consists of a single-frequency sinusoidal pulse for 
logic 1 and a different frequency sinusoidal pulse for logic 0. The FSK is used in many applications 
including paging and cordless systems. Figure 6.3 1(a) shows the binary sequence 10010 that generates 
a BFSK waveform. The FSK spectrum depends both on the symbol rate/ m and maximum frequency 
deviation A/. For a sufficiently small frequency deviation when A/ < 3.4/ m , the FSK spectrum 
shown in Figure 6.31(b) occupies less frequency bandwidth required for the receiver according to 
Eq. (6.36) than the ASK spectrum shown in Figure 6.28(b). This is because higher FSK frequency 
components decrease faster than the corresponding higher ASK frequency components. 

The simple way to implement BFSK is to use a crystal oscillator with a switching capacitance. 
Figure 6.32 show the circuit schematic of a crystal transistor-controlled FSK modulator where the 
capacitance C2 is turned on or off by the transistor in accordance with the binary sequence delivered to 
its input to transmit different frequencies. The frequency shift corresponding to the required BFSK can 
be controlled by proper selection of the capacitances Ci and Co - Typically, the frequency deviations 
of about few megahertz are possible for such an FSK modulator, because larger deviation reduces the 
resonator quality factor and thus increases the phase noise. 



DIGITAL MODULATION 289 




The BFSK signal can be represented as a sum of two OOK signals generated by two binary 
sequences ni and n 2 , as shown in Figure 6.33. In this case, logics 1 and 0 for binary sequences should 
alternate with each other when logic 1 for binary sequence n\ corresponds to logic 0 for binary 
sequence ni and vice versa. The BFSK can be provided using an HQ modulator with quadrature 
components of the carrier frequency f 0 when the / and Q inputs are driven by a low-frequency 
oscillator, set at the frequency offset / m . The output of the oscillator is split into quadrature components 
and applied to the modulator, thus generating an SSB output at the frequency /o — f m (or/o +f m , 
depending on the phase relationships of / and Q inputs). The low-frequency oscillator can be on- 
to-off keyed by a binary sequence to toggle between the fixed frequency f 0 representing a carrier 
feedthrough mode and the modulated frequency fo —fm representing an SSB mode. 

In m-ary frequency shift keying (MFSK), n bits are combined into a symbol before transmission, 
and each symbol is represented by a separate frequency of the signal. There can be a total of m 
different frequencies, with each change in frequency occurring every symbol period equal to fi-bit 
periods. For example, in quaternary or 4FSK, the data is transmitted in symbols of two bits, as shown 
in Figure 6.34, where each symbol is represented by a different frequency of the transmitted signal. 
The signal can have any of four frequencies depending on the type of symbol (00, 01, 10, or 1 1). The 
frequencies are spaced equidistant from each other and the spacing generally is even multiples of the 
symbol frequency / s . The MFSK signal requires less bandwidth compared to BFSK; however, more 
resolution is required in the receiver. 



6.5.3 Phase Shift Keying 

Phase shift keying (PSK) modulation scheme is used to transmit digital information by shifting the 
phase of a carrier among several discrete values. When a binary sequence is to be transmitted, the 
phase is usually switched between 0° and 180°, and the binary phase shift keying (BPSK) signal can 
be defined as 

{V 0 cos (co 0 t + 4> 0 ) for logic 1 
(6.65) 
— V 0 cos (a) 0 t + (f> 0 ) for logic 0 



290 MODULATION AND MODULATORS 






Binary 
sequence 



BFSK 
signal 




OOK 
signal 



OOK 
signal 



FIGURE 6.33 BFSK signal as sum of two OOK signals. 



where <p 0 is the initial phase. Hence, there are two possible locations on an /-axis of the HQ diagram, 
so a binary one or zero can be sent, and the symbol rate is one bit per symbol. Figure 6.35(a) shows 
the binary sequence 10010 that generates a BPSK waveform. The PSK spectrum generally contains 
the carrier and sideband components located symmetrically at offset frequencies multiple to a bit rate 
f m . However, for a phase keying of ± 180°, the carrier component is absent, and BPSK spectrum 
shown in Figure 6.35(b) becomes similar to an ASK spectrum with suppressed carrier, with zero 



DIGITAL MODULATION 



0 0 0 



Binary 
sequence 




FIGURE 6.34 Binary sequence and 4FSK modulation. 



(a) 



Binary 
sequence 



BI'SK 
signal 



Mf- 180° 



j_L 



0.5 



J__i 



A - 7f. A 5.L A - 3/ m fo -A jS +/• A + 3/ m /<, - 5/„ /, + Ifm f 

(*) 

FIGURE 6.35 Binary modulating signal, BPSK signal, and spectrum. 



292 MODULATION AND MODULATORS 



crossing corresponding to the even-order sidebands and increased level of the odd-order sideband 
components, both having approximately the same bandwidths. 
An m-ary phase shift keying (MPSK) signal can be defined as 

v(t) = V 0 cos ( a> 0 t + 2n h 4> 0 ] (6.66) 

V m ) 

where i — 1, 2, . . . , m. In MPSK, n bits are combined and transmitted as a symbol, with a total of 
m — 2" different symbols, and each symbol is a different combination of n bits. Each transmitted 
symbol is represented by a different phase of the signal. Therefore, the signal can have up to m phase 
changes occurring every symbol period. The transmitted signal in Eq. (6.66) can be rewritten using 
a trigonometric identity as 

v(t) = V 0 cos ( 27r — ) cos (ct) 0 t + (f> 0 ) — V 0 sin ( 27r — ] sin (co 0 t + <f> 0 ) . (6.67) 
\ m / \ m ) 

Since PSK requires coherent demodulation with phase reference, differential phase shift keying 
(DPSK) avoids the requirement for an accurate local oscillator phase by using the phase during the 
immediately preceding symbol interval as the phase reference [17]. As long as the preceding phase 
is received correctly, the phase reference is accurate. There are a variety of encoding techniques for 
implanting a DPSK system. One technique, which can be used for one-bit-at-a-time transmission, 
is to obtain a differential binary sequence from the input binary sequence and then assign phases to 
the bits in the differential sequence. The differential binary sequence is generated by repeating the 
preceding bit in the differential sequence if the message bit corresponds to logic 1 or by changing to 
the opposite bit if the message bit corresponds to logic 0. Phases are then assigned to the differential 
binary sequence by transmitting 0° for logic 1 and 1 80° for logic 0. Another common DPSK encoding 
scheme is to group bits into singles, pairs (dibits), or triples (tribits), and then associate a particular 
phase change with each group. 

In quadrature phase shift keying (QPSK) or 4PSK with n — 2 and m — 4, two bits are combined 
to form a symbol before transmission when each symbol is represented by a different phase of the 
transmitted signal and phase change occurs every symbol period. The phases are usually spaced 
equally so that the transmitted phases for 4PSK signal are 90° apart. Figure 6.36 shows the binary 




FIGURE 6.36 Binary sequence and 4PSK modulation. 



DIGITAL MODULATION 293 



sequence and 4PSK modulation according to Eq. (6.67) with (f> 0 — — 90°. Any sudden phase change 
results in non-constant envelope. The more abrupt the phase change, the more envelope changes and 
the wider the spectrum is, with the worst case of a 180° phase change. To minimize the power in 
spectral adjacent channels, pulse shaping is usually applied. In 8PSK, there are eight states of points 
spaced equally with 45°, 16PSK corresponds to the signal with equal phase spacing of 22.5°, and 
so forth. 

It is convenient to represent a PSK signal, as well as some other types of the modulated signal, 
in the form of a phasor or constellation diagram. Using this scheme, the phase of the signal is 
represented by the angle around the circle, and the amplitude by the distance from the origin or center 
of the circle. In this way, the signal can be resolved into quadrature components representing the 
sine for in-phase (/) component and the cosine for the quadrature (Q) component. Most PSK systems 
use constant amplitude, and therefore, points appear on one circle with constant amplitude and the 
changes in state being represented by movement around the circle. For a BPSK using phase reversals, 
the constellation diagram represents two points appearing at opposite points on the / axis, as shown 
in Figure 6.37(a). Other forms of PSK may use different points on the circle, and there can be more 
points on the circle. The constellation diagram for a QPSK signal with initial phase started at 45° is 
shown in Figure 6.31(b). Since a rotation of this constellation diagram has no effect on modulation 
performance, it can also start at zero phase, as shown in Figure 6.37(c). Figure 637(d) shows the 
trajectories for an 8PSK signal, where the bits are mapped to possible phases according to the Gray 
coding. In this case, the bits are numbered such that each adjacent phase means just bit difference so 
that when a phase mistake is made and the most likely one is the nearest phase, then only one bit is 
decoded incorrectly. Using a constellation view of the signal enables quick fault-finding in a system. 




FIGURE 6.37 Trajectories for (a) BPSK, (b) QPSK, (c) shifted QPSK, (d) 8PSK. 



294 MODULATION AND MODULATORS 



If the problem is related to phase, the constellation will spread around the circle. If the problem is 
related to magnitude, the constellation will spread off the circle, either towards or away from the 
origin. These graphical techniques assist in isolating problems much faster than when using other 
methods. 

A form of QPSK is used in the Digital Advanced Mobile Phone System (DAMPS) cellular 
architecture, as well as for the forward link from the base station to the handset in the IS-95 Code 
Division Multiple Access (CDMA) cellular system. On the reverse link from the handset to the 
base station, offset-quadrature phase shift keying (OQPSK) is used to prevent transitions through the 
origin. Figure 6.38 shows the block diagram of an OQPSK modulator where a serial bit stream is 
transformed by a serial-to-parallel (S/P) converter to separate / and Q bit streams. In OQPSK, one 
vector component is delayed by one bit period (one half of a symbol period), so the vector will move 
down first and then over, thus avoiding moving through the origin. In this case, the range of phase 
transitions is 0° and 90° when the possibility of a phase of 180° is eliminated and occurs twice as 
often, but with half intensity of the QPSK. If a QPSK modulated signal undergoes filtering to reduce 
the spectral side lobes, the resulting waveform will no longer have a constant envelope, and the 
occasional 1 80° shifts in phase will cause the envelope to go to zero momentarily. Moreover, a high 
nonlinearity of the active device transfer characteristic in this operation region close to pinch-off tends 
to introduce a significant nonlinear distortion in the signal transmission through the power amplifier. 
However, when an OQPSK signal undergoes bandlimiting, the resulting intersymbol interference 
causes the phase transition of ±90°, and the envelope will never go to zero. 

Another method to avoid the transitions through the origin is to use the differential phase shift 
keying (DPSK) system when the information is not carried by the absolute value, but is carried by the 
transitions between states. A DPSK transmission system can have transitions from any symbol position 
to any other symbol position. The 7r/4-DPSK modulation format where the carrier trajectory does 
not go through the origin is widely used in many wireless applications such as IS-54 North American 
Digital Cellular (NADC) standard, cordless Personal Handyphone System (PHS), or trunked radio 
system TETRA (Trans European Trunked Radio). It is based on two QPSK constellations, being 
offset between each other by 45°, where transitions must occur from one constellation to the other. 
The data are encoded in the magnitude and direction of the phase shift, but not in the absolute position 
on the constellation. 

This is achieved by using the two QPSK modulators running in parallel and using one symbol 
from each one each time (2r), as shown in Figure 6.39(a). From the signal constellation diagram 



/(rl 



Balanced 
modulator 



Serial 
bit stream 



S/P 
converter 



Combiner 



•v(f) 



Q(i) 



Balanced 
modulator 



sinca c / 

FIGURE 6.38 Block diagram of OQPSK modulator. 



DIGITAL MODULATION 295 



Serial 

bit stream 



An 



ST 

converter 



(Jin 



S l> 

converter 



at) 



Balanced 
modulator 



etisfOof 
sinuv 



Ha lanced 
modulator 



Balanced 
modulator 



sin (!) (J / 



Balanced 

modulator 



(a) 



Combiner 



Combiner 





0 / 0 

(*) (c) 
FIGURE 6.39 Block and constellation diagrams for tt/4-QPSK and 3tt/8-8PSK. 



shown in Figure 6.39(b), it is possible to obtain that the maximum phase change is 135°. The 3jt/8- 
8PSK modulation scheme is similar to 7T/4-DPSK in the sense that rotation of the constellation 
occurs from one time interval to the next. This time, however, the rotation of the constellation 
from one symbol to the next is 37r/8. This modulation scheme is used in the EDGE (Enhanced 
Data Rates for GSM Evolution) system and provides 3 bits per symbol. Its constellation diagram 
where the continuous rotation of the symbols by a 3tt/8 offset prevents the signal from crossing 
through the origin is shown in Figure 6.39(c). This condition can be viewed as having two 8PSK 
constellation planes offset by 3^/8 and swapping from one plane to the other at every consecutive 
symbol time. 

Generally, filtering allows the transmitted bandwidth to be significantly reduced without losing 
the content of the digital data. There are many different varieties of filtering techniques, among which 



296 



MODULATION AND MODULATORS 




(/>) DalaQ 



FIGURE 6.40 Circuit topologies of single-stage and balanced BPSK modulators. 

the most common are the raised-cosine, the square-root raised-cosine and the Gaussian filters [27]. 
The raised-cosine filters, being a class of Nyquist filters, have the property that their impulse response 
rings at the symbol rate. This means that time response of the filter goes through zero with a period that 
exactly corresponds to the symbol spacing. Nyquist filters heavily filter the signal without blurring the 
symbols together at the symbol times. This is important for transmitting information without errors 
caused by inter-symbol interference. Usually, the filter is split, with one half in the transmit path and 
the other half in the receiver path, being commonly called the square-root raised-cosine filters, so that 
their combined response is that of a Nyquist filter. The Gaussian filters used in GSM (Global System 
for Mobile communications or Global Speciale Mobile) represents a Gaussian shape in both the time 
and frequency domains, without ringing inherent in the raised-cosine filter. As a result, its effects in 
the time domain are relatively short and each symbol interacts significantly with only preceding and 
succeeding symbols. 

Accurate bi-phase modulation with equal amplitudes and 180° phase shift can be obtained in 
a reflection-type modulator if the switched device (p-i-n diode, MESFET, or HEMT) provides 
zero impedance when it is turned on and infinite impedance when it is turned off. Figure 6.40(a) 
shows the simplified schematic of a bi-phase BPSK modulator based on a hybrid quadrature Lange 
directional coupler and microwave MESFET devices in a switching-mode operation [28]. To make 
equal reflections from MESFETs in two states by minimizing the device parasitic capacitive and 
inductive elements, a thin-film resistor can be placed between drain and source terminals of each 
MESFET. To achieve a broadband modulator performance with a reflected power balance of ±1 dB 
and a bi-phase difference of 180° ±10° over the full octave range from 4 to 8 GHz, a tandem 
directional coupler can be used. A quadri-phase QPSK modulator can be designed using two bi- 
phase BPSK modulators, a 90° hybrid input splitter, and a 90° output hybrid combiner, as shown in 
Figure 6.40(b). Near perfect amplitude and phase performance in millimeter-wave applications can 
be achieved by using pHEMT devices acting as switches. In this case, a multifunctional HQ vector 
modulator for multilevel modulation schemes can be built using two quadri-phase QPSK modulators 
with a 90° hybrid input splitter and an output in-phase Wilkinson combiner [29]. 

6.5.4 Minimum Shift Keying 

The power spectral density of a PSK signal represents a sinc 2 (.t) spectrum with the main lobe cen- 
tered at the carrier frequency and side lobes of decreasing amplitudes extending on the neighboring 



DIGITAL MODULATION 297 



channels, the most important of which are adjacent and alternate, creating interference with these 
channels, which is called the spectral regrowth. Spectral regrowth can be reduced by using a modu- 
lation format called the minimum shift keying (MSK), which occurs when the phase changes of the 
transmitted signal are smooth as opposed to instantaneous changes in QPSK. Smooth phase changes 
in MSK are provided by filtering the digital data before modulation, thus resulting in the side lobes 
much lower in amplitude compared to BPSK or QPSK. However, the addition of a filter in the data 
path creates symbol distortion, and therefore, the choice of filtering must take into consideration the 
tradeoff between symbol distortion and side lobe suppression [30], 

MSK can be mathematically understood as an OQPSK with a special pulse shaping in such a 
way that the transitions between symbols are smoothed and there are no abrupt changes in phase or 
frequency. If sinusoidal pulses are employed instead of rectangular shapes, the modified signal can 
be defined as MSK and equals to 

v(t) = ai(/)Vocos y— t^j coscoot + aQ(t)Vo sin sincoo/ (6.68) 

where a\ and «q are two data streams consisting of even and odd bits, being separated from the binary 
input bit stream arriving at a rate of 1/t baud. Figure 6.41(a) shows the modified in-phase bit stream 
waveform with the corresponding amplitudes ± 1 shown inside waveforms. The in-phase carrier, 
representing by the first term in Eq. (6.68), is obtained by multiplying the waveform in Figure 6.41(a) 
by cosa>o?, as shown in Figure 6.41(£>). Similarly, the sinusoidally shaped odd-bit stream and the 
quadrature carrier, multiplied by sincoof, are shown in Figures 6.41(c) and 6.41(0"), respectively. The 
composite MSK signal u(f), the addition of Figures 6A\(b) and 6.41(d), is shown in Figure 6.41(e). 

To better understand the waveform shown in Figure 6.41(e), Eq. 6.68 can be rewritten using a 
well-known trigonometric identity as 

v(t) = V 0 cos [W + bk(t)—t + & J (6.69) 

where b k is equal to + 1 when ai and «q have opposite signs and b k is equal to — 1 when cij and «q 
have the same sign, and cf>^ is equal to 0 or it corresponding to a\ — + 1 or — 1 , respectively. Note that 
b k (i) can also be written as — « I (r)flQ(f). 

From Eq. (6.69) and Figure 6.41(e), the following basic properties of MSK can be derived: 



1. It has constant envelope. 

2. There is phase continuity in the RF carrier at the bit transition instants. 

3. The signal represents an FSK signal with signaling frequencies / + — fo + l/(4r) and /_ = 
fo — 1/(4t), where /o = a>ol(2it), with the frequency deviation equals to half the bit rate, that 
is, A/ =/+ — /_ = 1/(2t). This is the minimum frequency spacing that allows the two FSK 
signals to be coherently orthogonal. Since the frequency spacing is only half as much as the 
conventional 1/r spacing used in noncoherent detection of FSK signals, MSK is also referred 
to as the fast frequency shift keying (FFSK). 

4. The excess phase of the MSK signal, referenced to the carrier phase, is given by the term 

<p(t) = h{t)—t = ±—t 
2r 2t 

in Eq. (6.69), which increases or decreases linearly during each bit period of r seconds. A bit b k 
of + 1 corresponds to an increase of the carrier phase by 90° and corresponds to an FSK signal 
at the higher frequency / + . Similarly, b k — — 1 implies a linear decrease of phase by 90° over 
r seconds, corresponding to the lower frequency /_. In order to make the phase continuous at 
bit transitions, the carrier frequency f 0 should be chosen such that/ 0 is an integral multiple of 
1/(4t), one-fourth the bit rate. 



298 MODULATION AND MODULATORS 



I/O 



do 

L 

/ +' \ / 


(V- 

1 1 \ 


0[ CI,; 

5r It, 


2t 




T 




-1 /\ -1 / 


/ 


/ /\ \ / 


A x 


\ A A /V 


C/|l'/)COS 71 / COSf;) ( >f 

2t 


\\ / \ // \\ 


/ \ #/ \ 


\ / / \ \ / / 






ft- 1 




an(/)sm 51 r 
2t 



-J- 1, 



/ 


/V. / 
1 \\ / 

' \ \y\ 


/\\ /V / 


v\ //\ 
i % / / 1 


\ ' 2t 

— r— ^ 


\ \ J 

. I/-. 


V\\ 1 

y x 
■M. ■' 


y % 

i - A i 


/ V 
/- 

fe, 


, / / / 




FIGURE 6.41 Binary modulating signal and MSK signal. 



Thus, MSK can be viewed either as an OQPSK signal with cosine pulse weighting or as a 
continuous phase frequency shift keying (CPFSK) signal with a frequency separation equal to one-half 
the bit rate. Figure 6.42 shows the block diagram of an MSK modulator corresponding to Eq. (6.68). 

By using a Gaussian pulse shaping, the MSK spectral performance can be improved even further 
in terms of lower side lobes as a result of smoother transitions between symbols with optimum 
filter roll-off factor to compromise between a bit rate and out-of-band interference. The modulation 
obtained this way is called the Gaussian minimum shift keying (GMSK). In GMSK modulation, a 
data stream is passed through a Gaussian filter and the filtered response drives an HQ modulator, 



DIGITAL MODULATION 299 



Serial 
bit 3 L ream 



An 



S'P 
L'tm verier 



jr. f 
cns 7T 



Balanced 
modulator 



Balanced 
modulator 



eosiV 



Balanced 
modulator 



Balanced 
moduli* lor 



Cornbincr 



■no 



FIGURE 6.42 Block diagram of MSK modulator. 



with similar architecture as shown in Figure 6.42. GMSK can also be achieved by using a 7r/2-shift 
BPSK or CPFSK modulator driving the voltage-controlled oscillator (VCO) included in a phase- 
locked loop together with Gaussian filter [31,32]. GSMK is used in such communication systems 
as GSM850/900, DECT (Digital European Cordless Telephone), CDPD (Cellular Digital Packet 
Data), DCS1800 (Digital Communication System in the 1800 MHz band), or PCS1900 (Personal 
Communications Services in the 1900 MHz band). 



6.5.5 Quadrature Amplitude Modulation 

The above described digital modulation schemes use either amplitude or phase carrier changes to 
transmit binary data stream, with the constellation points lying on a circle of constant amplitude. 
However, much more information can be transmitted if both amplitude and phase are varied, as it can 
be done using quadrature amplitude modulation (QAM), based on a proper digital processing. An 
m-ary quadrature amplitude modulation (MQAM) signal can be defined as 

v(t) — V; cos (ci) 0 t + fa + 4>o) = Vi cos (fa) cos (a> 0 t + <f> 0 ) — V\ sin (fa) sin (w 0 t + <p 0 ) (6.70) 

where V; are the amplitudes, (j>\ are the phases, and (p 0 is the initial phase, ( = 1, 2, . . . , m [33,34], 
Figure 6.43(a) shows the block diagram of the MQAM modulator transmitting a bit stream with n 
bits and m symbols. The QAM constellation is produced by separately amplitude modulating the / 
and Q carriers, which have the same frequency but are 90° out of phase. The most common form 
of QAM is a square QAM, or a rectangular QAM with equal numbers of / and Q states, where the 
constellation points are usually arranged in a square grid with equal vertical and horizontal spacing. 

In 16-state QAM (16QAM), there are four / values and four Q values resulting in a total of 16 
possible states for the signal. It can transit from any state to any other state at every symbol time. Since 
16 = 2 4 , four bits per symbol can be sent, with symbol rate equal to one-fourth of the bit rate. Note 
that 2QAM is in fact BPSK, as well as 4QAM is the same as 4PSK or QPSK, which follows from 
the constellation diagrams shown in Figures 637(b) and 6.43(b). Also, the error-rate performance of 
8QAM is close to that of 16QAM, but its data rate is only three-quarters that of 16QAM. Therefore, 
the most common forms are 16QAM, 64QAM, 128QAM, and 256QAM. By moving to a higher order 
constellation, it is possible to transmit even more bits per symbol. However, the symbols become 
very close to each other, and are thus more subject to errors due to noise and distortion. Figure 
6.43(c) shows the constellation diagram for 16QAM with a Gray coded bit assignment. 64-QAM and 
256-QAM are often used in digital cable television and cable modem applications. 



300 MODULATION AND MODULATORS 



n - log>m 
bils 



Serial 
bit stream 



S;P 
convener 



Assign 
amplitude f, 
iinJ 
phase 0; 



I', sin<|) 



Ralanc-ed 
modulator 



iialanced 
imidulalor 



Com hi tier 



■ l'(') 



(a) 



01 



00 



10 



0011 0111 



0010 0110 



0000 


0100 


1 100 


1 ( 


• 


• 


• 


• 


0001 


0101 


1101 


1001 


• 


• 


• 


• 



mi ion 

• • 

1110 1010 



(b) (c) 

FIGURE 6.43 Block diagram of QAM modulator and constellation diagrams. 



6.5.6 Pulse Code Modulation 

Pulse code modulation (PCM) was originally developed in 1939 by Alec Reeves for voice com- 
munication as a method for transmitting digital signals over analog communications channels [35]. 
This is generally a method of transmitting continuous signals in which the amplitude of the signal is 
sampled regularly at uniform intervals and then quantized to a series of symbols in a numeric (usually 
binary) code. Pulse-code modulation is very popular because of many advantages it offers such as 
inexpensive digital circuitry required in the system, all-digital transmission, possible encryption with 
further digital processing, or error minimization by appropriate coding of the signals. PCM variants 
are based on different mathematical techniques for quantization, including linear, logarithmic, and 
adaptive. Quantization is the process of converting a continuous amplitude into one of a finite number 
of discrete amplitudes. 

Figure 6.44 shows how an analog signal in the form of a sine wave is converted (sampled) into 
16 amplitude levels with equal spacing by a 16-level quantizer. For each sample, one of the available 
values shown on the K-axis is chosen by some algorithm (in this case, the floor function is used when 
floor(x) is the largest integer not greater than x). This produces a fully discrete representation of the 
input signal (shaded area) that can be easily encoded as digital data for storage or manipulation. For 
this sine wave signal, we can verify that the quantized values at the sampling moments are 7, 9, 11, 
12, 13, 14, 14, 15, 15, 15, 14, and so on. Encoding these values as binary numbers would result in the 
following set of nibbles (or half-octets): 01 11, 1001, 1011, 1100, 1101, 1 1 10, 1 1 10, 1 1 1 1, 1 1 1 1, 1 1 1 1, 
1110, and so on. These digital values could then be further processed or analyzed by a purpose-specific 
digital signal processor or general purpose central processing unit (CPU). Several PCM data streams 



DIGITAL MODULATION 301 



A 



T 



\ 



1 



: 



3 



1^1 



J 



3 



Digil 


Binary 
code 


Gray 
code 


0 


(KKX) 


(KMX) 


1 


01)01 


HI 1(11 


2 


0010 


(Kill 


3 


0011 


0010 


4 


0100 


0110 


5 


0101 


0111 


6 


0110 


0101 


7 


0111 


01(H) 


8 


1000 


1100 


9 


1001 


1101 


10 


1010 


1111 


11 


1011 


1110 


12 


11(H) 


1010 


13 


1101 


1011 


14 


1110 


1001 


15 


1111 


1000 



FIGURE 6.44 Signal sampling and quantization for 4-bit PCM. 



could also be multiplexed into a larger aggregate data stream, generally for transmission of multiple 
streams over a single physical link. There are many ways to implement a real device that performs 
this task. In real systems, such a device is commonly implemented on a single integrated circuit that 
lacks only the clock necessary for sampling, and is generally referred to as an ADC (analog-to-digital 
converter). These devices will produce on their output a binary representation of the input whenever 
they are triggered by a clock signal, which would then be read by a processor of some sort. 

Generally, quantization reduces the degree of accuracy of representation of the sampled signal, 
which is characterized by a quantization error or noise. To evaluate the quantizer performance, 
one commonly used measure is the quantizer signal-to-noise ratio (SQNR). Assuming a uniform 
distribution of input signal values, the quantization noise is a uniformly-distributed random signal 
with peak-to-peak amplitude of one quantization level, making the amplitude ratio 2"/l. Then, if the 
input to the quantizer is a full-scale sine wave signal (that is, the quantizer is designed such that it has 
the same minimum and maximum values as the input signal), the quantization noise approximates a 
sawtooth wave with peak-to-peak amplitude of one quantization level and uniform distribution [17]. 
As a result, for a n-bit quantizer, 

SQNR(dB) = 20nlog 10 2 + 1.76 = 6.02n + 1.76 (6.71) 

Thus, for an 8-bit quantizer, the SQNR for a full-scaled sine wave is about 50 dB, and each extra 
quantization bit increases the SQNR by roughly 6 dB. In practical digital telephone systems, 256 = 
2 8 levels are used to keep quantization error to an acceptable level, whereas 65,536 = 2 16 levels are 
used for the CD digital system. 

If the quantized samples are transmitted directly over a channel, such a process is simply called the 
pulse amplitude modulation (PAM) process in the transmission system. In PCM system, the quantized 
signal is coded into a block of digits for transmission. The decimal-to-binary conversion can be done 
in various ways, for example, using two simple coding rules based on the natural binary and Gray 
codes, as shown in Figure 6.44 for a 16-level sample into 4 binary digits. In the Gray coding, adjacent 
levels differ by only a single bit, and hence, for equally likely bit errors in each position, a single 
channel bit error is more likely to produce an adjacent output level than in other codes. 

Some forms of PCM combine signal processing with coding. Differential pulse code modulation 
(DPCM) encodes the PCM values as differences between the current value and the predicted value. 



302 MODULATION AND MODULATORS 



An algorithm predicts the next sample based on the previous samples, and the encoder stores only 
the difference between this prediction and the actual value. If the prediction is reasonable, fewer bits 
can be used to represent the same information. For audio signals, this type of encoding reduces the 
number of bits required per sample by about 25% compared to PCM. Adaptive differential pulse 
code modulation (ADPCM) is a variant of DPCM that varies the size of the quantization step, to 
allow further reduction of the required bandwidth for a given SQNR. Delta modulation (DM or A- 
modulation), another variant, is the simplest form of DPCM where the difference between successive 
samples is encoded into n-bit data streams, and the transmitted data is reduced to a 1-bit data stream. 



6.6 CLASS-S MODULATOR 

A class-S modulator is a high-efficiency low-frequency power amplifier based on pulse-width mod- 
ulation (PWM), and its output is a baseband envelope signal including a dc component [5,36]. 
Pulse-width modulation (or pulse-duration modulation) uses a square-wave signal whose pulse width 
is modulated resulting in the variation of the average value of the waveform. Such a modulator 
produces only positive output current and, therefore, requires only a single transistor and a single 
diode that is required to provide a suitable reverse-direction path, as shown in Figure 6.45. The output 
voltage can have any value between nearly zero voltage and the supply voltage V cc . The low-pass 
filter (LPF) should have high impedance to the switching (sampling, or clock) frequency and its 
harmonics to provide a high level of spectral purity of the output RF signal. Usually, it is preferable 
to use a first-order LPF, since further increasing the number of its elements improves the modulator 
performance insignificantly, but significantly increases size and complexity of a Class-S modulator. 
When the transistor is turned on, the supply voltage V cc is connected to the input of the LPF, and the 
load current flows through the device. However, when transistor is turned off, the load current flows 
through the diode, and the voltage at the LPF input is equal to the voltage across the diode. 

In an ideal operation mode, the active device never experiences simultaneous nonzero voltage and 
nonzero current resulting in 100% efficiency. However, in practical implementation, the losses in the 
diode and transistor degrade efficiency due to the finite values of their on-resistances and finite rise 
and fall times, especially at higher operation frequencies. To minimize the intermodulation distortion 
of the envelope signal, it is necessary to choose correctly the LPF parameters for a maximally flat 
frequency response, which can be calculated from 

1 

L = 0.7— C= (6.72) 

tt/ c 2.8jr/ c J?L 

where f c is the cutoff frequency of the low-pass filter. 




FIGURE 6.45 Schematic of Class-S modulator. 



CLASS-S MODULATOR 303 




FIGURE 6.46 Waveforms of pulse-width modulation of envelope signal. 



Generally, the PWM signal can be accomplished by several techniques, the most popular of 
which is a comparator method shown in Figure 6.45. In this case, a comparison of the envelope 
input to a triangular reference wave shown in Figure 6.46(b) by a comparator produces a PWM 
switching signal, the width of which varies with envelope amplitude, as shown in Figure 6.46(c). 
The comparator produces maximum output when the input signal is larger than the triangular wave 
and zero output when the input signal is smaller than the triangular wave. The triangular wave can 
be supplied directly from a function-generator circuit or obtained by integration of the output of a 
switching generator with rectangular-wave signal, which is shown in Figure 6.46(a). The maximum 
modulation frequency depends strongly on the switching frequency required for a PWM process. To 
keep the spurious products in the output spectrum at least 30-40 dB below the carrier, the switching 
frequency must four or five times exceed the highest modulation frequency. In practice, to restore the 
input envelope with minimum level of intermodulation components using an LPF, as shown in Figure 
6.46(d), it is better to use a factor of 10. An efficiency of about 90% can be achieved for a Class-S 
modulator with an envelope bandwidth up to 150 kHz [36,37], However, a PWM is an inherently 
nonlinear process that generates intermodulation components with as higher level as wider RF signal 
bandwidth [38]. 



304 MODULATION AND MODULATORS 



Therefore, for the envelope-varying systems with wider bandwidths and stronger requirements for 
the intermodulation distortion levels, it is preferable to use a Class-S modulator based on a low-pass 
delta-sigma modulation ( A E -modulation or DSM) scheme. For the same switching frequency (or 
oversampling ratio), a system based on DSM can provide wider bandwidth and lower distortion 
than a PWM system. The signal in a delta-sigma modulator is digitized by a quantizer (single-bit 
comparator), the output of which is subtracted from the input signal through a digital feedback loop 
acting as a discrete-time sharp low-pass filter and quantization noise is forced outside of the band 
of interest [39,40]. The degree of suppression of the quantization noise depends on an oversampling 
ratio, which is the ratio of a sampling frequency to twice the bandwidth or highest frequency of the 
signal being sampled. 



6.7 MULTIPLE ACCESS TECHNIQUES 
6.7.1 Time and Frequency Division Multiplexing 

In the history of electrical communications, the earliest reason for sampling a signal was to interlace 
samples from different telegraphy sources, and transmit them over a single telegraph cable. The 
electrical engineer W. Miner used an electromechanical commutator for time-division multiplex of 
multiple telegraph signals in 1903, and also applied this technology to telephony where he obtained 
intelligible speech from channels sampled at a rate above 3500^1300 Hz by means of a pulse- 
amplitude modulation rather than PCM. Thus, time division multiplexing (TDM) is alternate process 
in which a number of signals or bit streams are transmitted simultaneously in one communication 
channel by letting the signals occupy the same frequency band on a time-sharing basis. In such 
systems the transmitter is switched or commutated to each signal channel sequentially. In this case, 
the time domain is divided into several recurrent timeslots of fixed length, one for each subchannel, 
when a sample byte or data block of subchannel 1 is transmitted during timeslot 1, subchannel 2 
during timeslot 2, and so on. The receiving system must then be switched in synchronism with the 
transmitter to separate the various signals prior to final demodulation. In circuit switched networks 
such as the Public Switched Telephone Network (PSTN) there exists the need to transmit multiple 
subscribers' calls along the same transmission medium, and to accomplish this, network designers 
make use of TDM that allows switches to create channels within a transmission stream. 

Frequency division multiplexing (FDM) is a form of signal multiplexing that involves assigning 
non-overlapping frequency ranges to different signals. Since the portion of the spectrum to be utilized 
is determined by the carrier frequency, different signals can modulate carriers of different frequencies 
and all of them can be transmitted simultaneously. The receiver can choose the desired signal band by 
means of selective filters. FDM can be also used to combine multiple signals before final modulation 
onto a carrier wave. In this case, the carrier signals are referred to as subcarriers. For example, in 
stereo FM transmission, a 38-kHz subcarrier is used to separate the left-right difference signal from 
the central left-right sum channel, prior to the frequency modulation of the composite signal. A 
television channel is divided into subcarrier frequencies for video, color, and audio. Digital Subscribe 
Line (DSL) uses different frequencies for voice and for upstream and downstream data transmission 
on the same conductors. 

Orthogonal frequency division multiplexing (OFDM) represents an FDM scheme utilized as a 
digital multicarrier modulation method where a large number of closely-spaced orthogonal subcarriers 
are used for data transmission [41]. In this case, the data is divided into several parallel data streams 
or channels, one for each subcarrier, and the subcarrier is modulated with a conventional modulation 
scheme such as QAM or PSK at a low symbol rate, maintaining total data rates similar to conventional 
single-carrier modulation schemes in the same bandwidth. OFDM has developed into a popular 
scheme for wideband digital communication used in applications such as digital television and audio 
broadcasting, wireless networking and broadband internet access. In OFDM, the subcarriers are 
chosen so that they are orthogonal to each other, meaning that cross-talk between the subchannels is 



MULTIPLE ACCESS TECHNIQUES 305 



eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the 
transmitter and the receiver when a separate filter for each subchannel is not required. As a result, an 
OFDM carrier signal is the sum of a number of orthogonal subcarriers, with baseband data on each 
subcarrier being independently modulated commonly using some type of QAM or PSK. 

6.7.2 Frequency Division Multiple Access 

When FDM is used as to allow multiple users to share a physical communication channel, it is 
called the frequency division multiple access (FDMA), which is the traditional way of separating 
radio signals from different transmitters. FDMA gives users an individual allocation of one or several 
frequency bands, allowing them to utilize the allocated radio spectrum without interfering with each 
other and coordinate access between multiple users. In FDMA, a predetermined frequency band is 
available for the entire period of communication, with stream data that may not be packetized. In 
the commercial radio broadcast bands, 535-1705 kHz for AM and 88-108 MHz for FM, each local 
broadcast station (user) is assigned to a specific slice of spectrum within the frequency band allocated 
for that purpose. As long as the station broadcasts, no other radio station in the same area can use that 
radio frequency bandwidth to send a signal. Another broadcast station can use that same bandwidth 
only when the distance between the stations is sufficient to reduce the risk of interference. Therefore, 
FDMA requires high-performing filters in the radio systems. In addition, because adjacent channel 
interference is an important factor in channel quality, frequency planning is a key consideration when 
selecting fixed or base station locations. 

It is important to distinguish between FDMA and frequency division duplexing (FDD). While 
FDMA allows multiple users simultaneous access to a certain system, FDD refers to how the radio 
channel is shared between the uplink and downlink (for instance, the traffic going back and forth 
between a mobile phone and a base station). Similarly, FDM should not be confused with FDMA. The 
former is a physical layer technique that combines and transmits low-bandwidth channels through a 
high-bandwidth channel. FDMA, on the other hand, is an access method in the data link layer. FDMA 
also supports demand assignment in addition to fixed assignment. Demand assignment allows all users 
apparently continuous access of the radio spectrum by assigning carrier frequencies on a temporary 
basis using a statistical assignment process. 

Orthogonal frequency division multiple access (OFDMA) is a multi-user version of the OFDM 
digital modulation scheme. Multiple access is achieved in OFDMA by assigning subsets of subcarriers 
to individual users. This allows simultaneous low data rate transmission from several users. 

6.7.3 Time Division Multiple Access 

Time division multiple access (TDMA) represents a channel access method for shared radio networks. 
TDMA systems can improve spectrum efficiency compared to FDMA because they allow not only 
assigning users into an available pair of channels, but also allowing several users to share the same 
frequency channel by dividing into different timeslots. The users transmit in rapid succession, one 
after the other, each using own timeslot. This allows multiple stations to share the same radio channel, 
while using only a part of its channel capacity. However, the user can only send or receive information 
at that time, regardless of the availability of other timeslots. Information flow is not continuous for any 
user, but rather is sent and received in bursts that are then reassembled at the receiving end and appear 
to provide continuous signal due to fast process. TDMA is used in the digital cellular systems such 
as GSM or DECT (Digital Enhanced Cordless Telecommunications) standard for portable phones, as 
well as in the satellite systems. Figure 6.47 shows the TDMA frame structure based on a data stream 
divided into frames with those frames divided into timeslots. 

TDMA is a type of TDM when multiple transmitters are used instead of a single transmitter 
connected to a single receiver. Because TDMA systems also split an allotted portion of the frequency 
spectrum into smaller slots (channels), they require the same level of frequency planning as FDMA 
systems. Generally, FDMA can provide extended battery life and talk time, more efficient use by 



306 MODULATION AND MODULATORS 













Guard periods (optional) 



Data stream di\ ided 
into frames 



Frames di\ ided into 
time slots. Fach user is 
allocated one slot 



Time slots contain data w ith a 
guard period if needed for 
n\ iKhroni/alion 



FIGURE 6.47 TDMA frame structure. 



accommodating more users in the same spectrum space, efficient utilization of hierarchical cell 
structures from macrocells to picocells and can efficiently handle video and audio data. However, 
in TDMA systems, the network and spectrum planning are intensive; they create interference at 
a frequency that is directly connected to the timeslot length; the frequency guard bands limit the 
potential bandwidth of a TDMA channel; and dropped calls are possible when users switch in and 
out of different cells. Handsets that are moving will need to constantly adjust their timings to ensure 
their transmission is received at precisely the right time, because as they move further from the base 
station, their signal will take longer to arrive, which also means that the major TDMA systems have 
hard limits on cell sizes in terms of range. 

In the GSM system, the synchronization of the mobile phones is achieved by sending timing 
advance commands from the base station that instructs the mobile phone to transmit earlier and by 
how much. The mobile phone is not allowed to transmit for its entire timeslot, but there is a guard 
interval at the end of each timeslot. As the transmission moves into the guard period, the mobile 
network adjusts the timing advance to synchronize the transmission. 



6.7.4 Code Division Multiple Access 

Code division multiple access (CDMA) is a class of modulation that uses specialized codes to 
provide multiple communication channels in a designated frequency range [42]. It is based on a 
spread spectrum technique used to increase spectrum efficiency over current FDMA and TDMA 
systems. In this case, CDMA spreads the information contained in a signal over the entire available 
bandwidth and not simply through one frequency. Due to the wide bandwidth of a spread-spectrum 
signal, it is very difficult to cause jamming, difficult to interfere with, and difficult to identify. CDMA 
systems are widely used in military and cellular applications. 

The use of CDMA for terrestrial cellular communications was first conceived in 1985 and was 
then standardized as IS-95 to become the first commercialized CDMA system. In North American 
cellular system, CDMA starts with a basic rate of 9600 b/s, and then it spreads to a transmitted bit 
rate, or chip rate, of 1.2288 MHz [43]. Spreading consists of applying digital codes to the data bits 
that increase the data rate while adding redundancy to the system. The chips (transmitted bits) are 
transmitted using a form of a QPSK modulation that is filtered to limit the bandwidth of the signal. 
When the signal is received, the coding is removed from the desired signal, returning it to a rate 
of 9600 b/s. The ratio of transmitted bits or chips to data bits is the coding gain, which is equal to 



MULTIPLE ACCESS TECHNIQUES 307 



128 or 21 dB and because of which an interference of up to 18 dB above the signal level (3 dB below 
the signal strength after coding gain) can be tolerated. The CDMA codes are designed to have very 
low cross-correlation. 

The basic differences of CDMA communication systems from the others are: 

1. Multiple users share one frequency. In a fully loaded CDMA system, there are about 35 
users on each carrier frequency. (Note that there are actually two carrier frequencies per 
channel, 45 MHz away from each other. One is for the base-mobile link, when it is called 
the forward direction, while the other is for the mobile-base link, which is called the reverse 
direction). 

2. The channel is defined by a code. There is a carrier frequency assignment, with a frequency 
bandwidth of 1.25 MHz. 

3. The capacity limit is soft. Additional users add more interference to the system, which can 
cause a higher data error rate for all users, but this limit is not set by the number of physical 
channels. 

CDMA makes use of multiple forms of diversity such as spatial, frequency, and time diversities. 
The traditional form of spatial diversity, using multiple antennas, is used for the cell site receiver. 
Another form of spatial diversity called the soft handoff is used during the process of handling off a call 
from one cell to the next. Soft handoffs allow the mobile telephone to communicate simultaneously 
with two or more cells, and the best signal quality is selected until the handoff is complete. Frequency 
diversity is provided in the bandwidth of the transmitted signal. With time diversity for multipath 
signals, the multiple correlative receiver elements can be assigned to different time-delayed copies 
of the same signal when the time-diverse signals are combined in optimal manner. One important 
aspect of CDMA is the use of Walsh codes based on the Walsh matrix, which is a square matrix with 
binary elements that always has a dimension that is power of two. 

Each user in a CDMA system uses a different code to modulate their signal. Choosing the codes 
used to modulate the signal is very important in the performance of CDMA systems. The best 
performance will occur when there is good separation between the signal of a desired user and the 
signals of other users. The separation of the signals is made by correlating the received signal with 
the locally generated code of the desired user. If the signal matches the desired user's code, then the 
correlation function will be high, and the system can extract that signal. If the desired user's code 
has nothing in common with the signal, the correlation should be as close to zero as possible (thus 
eliminating the signal), which is referred to as cross-correlation. If the code is correlated with the 
signal at any time offset other than zero, the correlation should be as close to zero as possible. This 
is referred to as auto-correlation and is used to reject multipath interference. 

In general, CDMA belongs to two basic categories: synchronous (orthogonal codes) and asyn- 
chronous (pseudorandom codes). Most modulation schemes try to minimize the bandwidth of the 
transmitted signal since bandwidth is a limited resource. However, spread-spectrum techniques use a 
transmission bandwidth that is several orders of magnitude greater than the minimum required signal 
bandwidth. One of the initial reasons for doing this was military applications, including guidance 
and communication systems. These systems were designed using spread spectrum because of its 
security and resistance to jamming. Asynchronous CDMA has some level of privacy built in because 
the signal is spread using a pseudorandom code; this code makes the spread-spectrum signals appear 
random or have noise-like properties. CDMA can also effectively reject narrowband interference. 
Since narrowband interference affects only a small portion of the spread-spectrum signal, it can 
easily be removed through notch filtering without much loss of information. Convolution encoding 
and interleaving can be used to assist in recovering this lost data. CDMA signals are also resistant to 
multipath fading. Since the spread-spectrum signal occupies a large bandwidth, only a small portion 
of this will undergo fading due to multipath at any given time. Like the narrowband interference, this 
will result in only a small loss of data and can be overcome. 



308 MODULATION AND MODULATORS 



Frequency reuse is the ability to reuse the same radio channel frequency at other cell sites within a 
cellular system. In the FDMA and TDMA systems, frequency planning is an important consideration. 
The frequencies used in different cells need to be planned carefully in order to ensure that the signals 
from different cells do not interfere with each other. In a CDMA system, the same frequency can be 
used in every cell because channelization is done using the pseudorandom codes. Reusing the same 
frequency in every cell eliminates the need for frequency planning in a CDMA system; however, 
planning of the different pseudorandom sequences must be done to ensure that the received signal 
from one cell does not correlate with the signal from a nearby cell. 

WCDMA (Wideband Code Division Multiple Access) is a wideband spread-spectrum channel 
access method providing higher speeds and support more users compared to CDMA and TDMA 
schemes and represents an air interface implemented in 3G (Third Generation) mobile telecommuni- 
cations networks [44]. The structures of the WCDMA physical channels, as well as the modulation 
and spreading techniques, differs in some respects from that of IS-95 CDMA. In WCDMA sys- 
tems, the QPSK nodulation is employed, with baseband filters representing the root raised-cosine 
filters having a roll-off factor of 0.22. WCDMA standards define a wide variety of data and chip 
rates. In this case, data rates for the traffic channels are between 32 and 1024 kSymbols/s. Chip 
rate of 3.84 MChips/s is chosen for a channel bandwidth of 5 MHz, with frame length of 10 ms 
and each frame is divided into 15 slots. A strong difference in the WCDMA structure with respect 
to IS-95 is the use of a pilot signal. This is a special data pattern that is identical for all code 
channels and is transmitted every 0.625 ms. WCDMA traffic channel structure is based on a single 
code transmission for small data rates and multicode for higher data rates. WCDMA supports FDD 
and TDD modes, using one carrier frequency for uplink and one for downlink and fifteen slots per 
radio frame in UMTS (Universal Mobile Telecommunications System) for time division. Multiuser 
detection and smart antennas can be used to increase capacity and coverage. Modulation schemes 
such as OQPSK or OMSK do not prevent zero-crossing for multiple channels at different ampli- 
tude levels with complex scrambling in 3G spread-spectrum systems. In this case, HPSK (Hybrid 
Phased Shift Keying), representing a complex spreading modulation technique that is based on Walsh 
coding and specific orthogonal spreading functions, can minimize zero-crossing and 0° phase shift 
transitions, thus improving the peak-to-average power ratio of the signal by approximately 1.0 to 
1.5 dB [45]. 



REFERENCES 

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Sept. 1901). 

2. H. Roder, "Amplitude, Phase, and Frequency Modulation," Proc. IRE, vol. 19, pp. 2145-2176, Dec. 1931. 

3. F. E. Terman, Electronic and Radio Engineering, New York: McGraw-Hill, 1955. 

4. H. S. Black, Modulation Theory, New York: Van Nostrand, 1953. 

5. H. L. Krauss, C. W. Bostian, and F. H. Raab, Solid State Radio Engineering, New York: John Wiley & Sons, 
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6. J. R. Carson, "Method and Means for Signaling with High Frequency Waves," U.S. Patent 1,449,382, Mar. 
1923 (filed Dec. 1915). 

7. A. A. Oswald, "Early History of Single-Sideband Transmission," Proc. IRE, vol. 44, pp. 1676-1679, Dec. 
1956. 

8. R. V. L. Hartley, "Modulation System," U.S. Patent 1,666,206, Apr. 1928 (filed Jan. 1925). 

9. O. G. Villard, "A High-Level Single-Sideband Transmitter," Proc. IRE, vol. 36, pp. 1419-1425, Nov. 1948. 

10. S. D. Bedrosian, "Normalized Design of 90 Degree Phase Difference Networks," IEEE Trans. Circuit Theory, 
vol. CT-7, pp. 128-136, June 1960. 

11. D. K. Weaver, "A Third Method of Generation and Detection of Single-Sideband Signals," Proc. IRE, vol. 
44, pp. 1703-1705, Dec. 1956. 



REFERENCES 309 



12. S. D. Thompson and A. M. Pavio, "A Monolithic Double Balanced Single Sideband Modulator," 1987 IEEE 
MTT-S Int. Microwave Symp. Dig., vol. 2, pp. 899-902. 

13. S. J. Parisi, "Monolithic, Lumped Element, Single Sideband Modulator," 1992 IEEE MTT-S Int. Microwave 
Symp. Dig., vol. 2, pp. 1047-1050. 

14. R. A. Heising, "Modulation Methods," Proc. IRE, vol. 50, pp. 896-901, May 1962. 

15. J. R. Carson, "Notes on the Theory of Modulation," Proc. IRE, vol. 10, pp. 57-67, Feb. 1922; reprinted in 
Proc. IEEE, vol. 51, pp. 893-896, June 1963. 

16. E. H. Armstrong, "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency 
Modulation," Proc. IRE, vol. 24, pp. 689-740, May 1936; reprinted in Proc. IEEE, vol. 52, pp. 1042-1062, 
Aug. 1964. 

17. J. D. Gibson, Principles of Digital and Analog Communications, New York: Prentice Hall, 1993. 

18. E. I. Manaev, "About Bandwidth to Receive Frequency-Modulated Signals without Nonlinear Distortions (in 
Russian)," Radioteklmika, vol. 3, pp. 54-61, May 1948. 

19. A. Grebennikov, RP and Microwave Transistor Oscillator Design, London: John Wiley & Sons, 2007. 

20. S. J. Lipoff, "Linearization of Direct FM Frequency Modulators," IEEE Trans. Vehicular TechnoL, vol. VT-27, 
pp. 7-17, Feb. 1978. 

21. E. Marazzi and V. Rizzoli, "The Design of Linearizing Networks for High-Power Varactor-Tuned Frequency 
Modulators," IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp. 767-773, July 1980. 

22. K. I. Kukk and V. G. Sokolinsky, Transmitter Devices of Multichannel Radio Relay Systems (in Russian), 
Moskva: Svyaz, 1968. 

23. R. V. Garver, "360° Varactor Linear Phase Modulator," IEEE Trans. Microwave Tlieoiy Tech., vol. MTT-17, 
pp. 137-147, Mar. 1969. 

24. T. Morawaki, J. Zborowska, and P. Miazga, "Multi-Octave Phase Modulators," 1984 IEEE MTT-S Int. 
Microwave Symp. Dig., pp. 378-380. 

25. F. Ali and N. Mysoor, "An Analog MMIC Phase Modulator for X-Band Satellite Transponder Applications," 
IEEE Microwave and Guided Wave Lett, vol. 2, pp. 445^46, Nov. 1992. 

26. Y. Tarusawa, H. Ogawa, and T. Hirota, "A New Constant-Resistance ASK Modulator Using Double-Sided 
MIC," IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 819-822, Sept. 1987. 

27. Application Note 1298, Digital Modulation in Communication Systems - An Introduction, Agilent Technolo- 
gies, 2001. 

28. B. E. Dobratz, N. J. Ho, G. E. Lee, and H. T. Suyematsu, "Microwave Analog and Digital Processors," 7979 
IEEE Int. Solid-State Conf. Dig., pp. 76-77. 

29. A. E. Ashtiani, S.-I. Nam, A. d'Espona, S. Lucyszyn, and I. D. Robertson, "Direct Multilevel Carrier 
Modulation Using Millimeter-Wave Balanced Vector Modulators," IEEE Trans. Microwave Theory Tech., 
vol. MTT-46, pp. 2611-2619, Dec. 1998. 

30. S. Pasupathy, "Minimum Shift Keying: A Spectrally Efficient Modulation," IEEE Commun. Mag., vol. 17, 
pp. 14-22, July 1979. 

31. K. Murota and K. Hirade, "GMSK Modulation for Digital Radio Telephony," IEEE Trans. Commun., vol. 
COM-29, pp. 1044-1050, July 1981. 

32. D. M. Klymyshyn, S. Kumar, and A. Mohammadi, "Direct GMSK Modulation with a Phased-Locked Power 
Oscillator," IEEE Trans. Vehicular TechnoL, vol. VT-48, pp. 1616-1625, Sept. 1999. 

33. T. Noguchi, Y. Daido, and J. A. Nossek, "Modulation Techniques for Microwave Digital Radio," IEEE 
Commun. Mag., vol. 24, pp. 21-30, Oct. 1986. 

34. F. Xiong, Digital Modulation Techniques, Norwood: Artech House, 2006. 

35. W. M. Waggener, Pulse Code Modulation Techniques, Cambridge: Springer, 2007. 

36. F. H. Raab and D. J. Rupp, "Class-S High-Efficiency Amplitude Modulator," RF Design, vol. 17, pp. 3-11, 
May 1994. 

37. F. H. Raab, B. E. Sigmon, R. G. Myers, andR. M. Jackson, "L-Band Transmitter Using Kahn EER Technique," 
IEEE Trans. Microwave Theory Tech., vol. MTT-46, pp. 2220-2225, Dec. 1996. 

38. F. H. Raab, "Intermodulation Distortion in Kahn-Technique Transmitters," IEEE Trans. Microwave Theory 
Tech., vol. MTT-44, pp. 2273-2278, Dec. 1996. 



310 MODULATION AND MODULATORS 



39. I. Galton, "Delta-Sigma Data Conversion in Wireless Transceivers," IEEE Trans. Microwave Theory Tech., 
vol. MTT-50, pp. 302-315, Jan. 2002. 

40. J. Choi, J. Yim, J. Yang, J. Kim, J. Cha, D. Kang, D. Kim, and B . Kim, "A AS -Digitized Polar RF Transmitter," 
IEEE Trans. Microwave Theory Tech., vol. MTT-55, pp. 2679-2690, Dec. 2007. 

41 . L. Litwin and M. Pugel, "The Principles of OFDM," RF Design, vol. 24, pp. 30-48, Jan. 2001. 

42. A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication, Boston: Addison-Wesley, 1995. 

43. D. P. Whipple, "The CDMA Standard," Applied Microw. & Wireless, vol. 6, pp. 24-37, Winter/Spring 1994. 

44. K.-D. Tiepermann, "Understand the Basics of WCDMA Signal Generation," Microwaves & RF, vol. 37, pp. 
83-86, Dec. 1998. 

45. Application Note 1335, HPSK Spreading for 3G, Agilent Technologies, 2000. 




Mixers and Multipli 



lers 



This chapter begins with a basic theory describing the operational principles of frequency conversion 
in receivers and transmitters. The different types of mixers, from the simplest based on a single 
diode to a balanced and double-balanced type based on both diodes and transistors, are described and 
analyzed. The special case is a mixer based on a dual-gate transistor that provides better isolation 
between signal paths and simple implementation. The frequency multipliers that historically were a 
very important part of the vacuum-tube transmitters can extend the operating frequency range. 

7.1 BASIC THEORY 

In wireless transmitters, it is generally required to convert a low-frequency baseband information 
signal to high frequency or frequencies at which the resulting signal can be effectively transmitted via 
antenna by electromagnetic propagation to the desired destination. This can be done either by a direct 
modulation of the high-frequency signal source or using a frequency converter traditionally called 
the mixer. Generally, any device (diode, vacuum tube, or transistor) that exhibits amplitude-nonlinear 
behavior can serve as a mixer, as nonlinear distortion results in the production of frequencies not 
present in the input. However, mixer as itself is fundamentally a linear device, which is shifting a signal 
from one frequency to another, keeping the properties of the initial signal (amplitude and phase), 
thus generally providing a linear operation. At the same time, mixers have relatively high levels of 
intermodulation distortion, spurious responses, and other undesirable nonlinear phenomena. Although 
mixers are equally important in wireless transmission and reception, traditional mixer terminology 
favors the receiving case because mixing was first applied as such in receiving applications [1,2]. 

The basic characteristic of a frequency-converter stage is its conversion transconductance, which 
is determined by the device transconductance of the input-electrode voltage to the output-electrode 
current. The general analysis of a frequency converter or mixer is applicable to all types of mixers no 
matter how or to what electrodes the oscillator and signal voltages are applied. Under assumption that 
the signal voltage is very small and the local oscillator (LO) voltage is large, the transconductance 
g m of the three-element device (or conductance g of the two-element device) may be considered as 
a periodically varying function of the oscillator sinusoidal voltage only, which can be written as a 
Fourier series 



CO 




(7.1) 



where coq is the angular frequency of the LO, n is the harmonic number, 




(7.2) 



i) 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



311 



312 MIXERS AND MULTIPLIERS 



On = — f g m cos rui) 0 t d (co 0 t). (7.3) 

Tt J 

0 

Use of the cosine series implies that the device transconductance is a single-valued function of the 
oscillator voltage that varies as coswo?. When a small signal u s = V s sina> s ? is applied to the device 
input, the resulting output current i in a first-order approximation with the oscillator-frequency terms 
only may be written as 

oo 

= g m V s sin aiof = a 0 V s s'ma> s t + V s ^^a n sinoV cos«cd 0 / = a 0 V s sina> s f 

n=\ 

v 00 v 00 

+ y XX sin(t<; s + na) Q )t + y X fl " sin(a; s - nm 0 )t. (7.4) 

n=\ n= 1 

Examination of Eq. (7.4) shows that the output current contains a component having a sum 
frequency (a> s + co 0 ) and a magnitude VJ2. The term fl[/2 is called the conversion transconductance 
g c . It is analogous to the device transconductance g m representing the factor that, when multiplied by 
the amplitude of the applied signal, will give the amplitude of the sum-frequency component of the 
output current. The conversion transconductance g c for the sum frequency is seen from Eq. (7.3) to 
have the value 



^ J 8m cos co 0 t d ((Dot). 



(7.5) 



In a common case, if a circuit tuned to the intermediate frequency (&> s — nco 0 ) is inserted in the 
output port, then the mixer operates as a downconverter converting the incoming signal frequency 
to a lower intermediate frequency. However, if a circuit inserted in the output port is tuned to 
the intermediate frequency (a> s + na> 0 ), then the mixer operates as an upconverter converting the 
incoming signal frequency to a higher intermediate frequency. For a given small-signal frequency w s , 
an ideal mixer with a perfect LO with no harmonics and no noise sidebands would produce only two 
intermediate-frequency outputs: one at the sum frequency (co s + co 0 ) and another at the frequency 
difference (co s — coq). Filtering can be used to select the desired intermediate-frequency product and 
reject the unwanted one, which is sometimes referred to as the intermediate-frequency (IF) image. 
Since n is an integer, it is evident that the intermediate frequency, in general, may be chosen to 
be the difference between the signal frequency and any integral multiple of the LO frequency. The 
conversion transconductance at the nth harmonic of the LO is given by 

Iaj s ±nai 0 _ 

gm = = 2 (7 ' 6) 



where /^in^ is the amplitude of the intermediate-frequency output current component (a> s ± nco 0 ). 

All mixers are multipliers in the sense that the various new output components they produce 
can be described mathematically as the multiplicative products of their input frequencies. Mixing is 
achieved by the application of two signals to a nonlinear device whose transfer characteristic may 
differ depending upon the particular device. An active device with square-law transfer characteristic 
is ideal for mixer performance, since the least number of undesired frequencies is produced. In this 
case, if a device has the transfer characteristic 



i(t) = a 0 v(t) + aiv 2 {t) 



(7.7) 



SINGLE-DIODE MIXERS 313 



then, with a two-tone input signal 



V(t) = Vi COS (Mi? + V 2 COS £02? 



(7.8) 



the output current becomes 

i(t) = a Q Vi cosa>it + a 0 V 2 cos co 2 t + <7i V[ 2 cos 2 a>if + aiV 2 2 cos 2 a> 2 t + 2aiViV 2 cosa>it cosco 2 ? 



where the first two terms are not necessary for mixer performance and must be filtered out in practical 
circuit. By the use of the trigonometric identity when 



the third and fourth terms are seen to represent a dc component and second harmonics of the input 
frequencies. Consequently, the final term in Eq. (7.9), which is called the product term, represents 
the desired output spectral components derived as 



where the amplitudes of the sum- and difference-frequency components are proportional to the 
product ViV 2 of the input-signal amplitudes. The sum and difference frequencies generated by the 
squared term in Eq. (7.7) are called the second-order intermodulation products, whereas the spectral 
frequency components generated by the cubed term such as 2a>i — co 2 and 2co 2 — coi are called the 
third-order intermodulation products. 

In a heterodyne receiver application, a low-level RF signal and an RF LO signal are mixed together 
to produce an intermediate difference frequency / IF =/rf — /lo (low-side conversion) or/ IF =/lo — 
/rf (high-side conversion) and a much higher sum frequency /'rf +/lo that is filtered out, as shown 
in Figure 7.1(a). In a particular transmitter operation, an RF LO signal and IF signal can be mixed 
together to produce an output RF signal / RF =/lo +/if (sum mixer) or/Rp =/lo — /if (difference 
mixer) with corresponding filtering, as shown in Figure 1.1(b). A single LO can provide the mixing 
function in the transmitter (upconversion) and receiver (downconversion), when both are used in 
radar or transceiver system. 

7.2 SINGLE-DIODE MIXERS 

The simplest mixer type that was for many years a key element in receiving systems was a diode mixer. 
Although other semiconductor junctions, such as the p—n junction, also exhibit nonlinear behavior, the 
metal-semiconductor Schottky-barrier diodes are widely used in modern mixers because of inherently 
low junction capacitance and high switching speed. As a result, Schottky diodes operate well into 
the millimeter-wave frequency with cutoff frequencies exceeding 200 GHz due to their much higher 
carrier mobility. Despite the fact that GaAs diodes are more expensive than those in silicon, they can 
provide better conversion loss and noise performance, especially at high frequencies. Besides, GaAs 
devices generally have higher breakdown voltages than silicon ones and better resistance to ionizing 
radiation. However, their advantage at lower frequencies is minimal and their cost is much greater. 

At low frequencies, the relationship between the current and voltage v across the mixer diode 
can generally be written as 



(7.9) 



a\ V 2 cos 2 a>{t — — aiV 2 (1 + cos2ci>i0 



(7.10) 



2aiVyV 2 coscoi? cos u> 2 t = a{ViV 2 [cos (a> 1 — u> 2 ) + cos (&>i + a> 2 )] 



(7.11) 




(7.12) 



314 MIXERS AND MULTIPLIERS 



Anicmia 



Low -noise [p 
amplifier Mixer low-pass filter 




/rf /lo 
■ f " = /lo /rf 



fix-, 



Mi\tr 



-Cg> 



Local oscillator 



RP Power 
bandpass filter amplifier 



Antenna 

w 



Ao+./ir 


n 










Ao +./IF 

/ Rr ,/i.o -./if 



Local oscillator 



(b) 



FIGURE 7.1 Frequency conversion in receiver and transmitter. 



where 7 sat is the reverse saturation current, Vj is the thermal voltage, and n is the ideality factor. The 
diode differential conductance at the operating point is therefore given by 

di / 8a t / v \ i +/ sat „ i . ^ , ,„ ... 

g = -r = exp I — I = — — = — i » / sat (7.13) 

If a cosine oscillator voltage is assumed to apply to the mixer diode, 

v = Vq -\- V cos ojqT (7.14) 

where Vo is the dc bias voltage, then its substitution into Eq. (7.12) allows us to rewrite the diode 
conductance g as a time- varying function by 

g(0 = -rr ex P -rr exp i7 — I — ~~ rr ex P ( ~~ rr I 2^ /k ~T7~ cos W (7.15) 

nV-r \nVi ) \ nVj ) nVj \nV T J f-J \nVj J 

where /k[W(nVr)] are the modified Bessel functions of the first kind of order k with argument 
W(nV T ) [3]. 



SINGLE-DIODE MIXERS 315 



If there were no charge storage in the diode, it would be infinitely fast because of absence any 
charge inertia, and the diode current could be changed in zero time instantly. However, there are two 
forms of charge storage: charge storage in the depletion region due to dopant concentration resulting 
in a junction capacitance Cj and charge storage due to minority-carrier charges injected into the 
neutral regions (holes stored in the /j-type region and electrons stored in the p-type region) resulting 
in a diffusion capacitance Cdiff- Thus, the total diode charge-storage capacitance for forward-bias 
voltages V 0 < 0.5ip is defined as 

C(v) = C m (v) + C s (v) = t— + C ] (V 0 ) " (7.16) 

dv \<p — v/ 

where y is the junction grading coefficient, r is the transit time of the diode, and (p is the junction 
potential [4]. 

When the diode has finite series resistance ; s , then both the forward and the reverse impedance 
at microwave frequencies are affected. In the forward direction, increasing the pumped voltage 
amplitude V across the diode junction causes the diode conductance to rise until it finally reaches 
maximum conductance g max = l/r s , as shown in Figure 7.2. In this case, the diode capacitance is 
represented by the diffusion capacitance C^g as a linear function of current i and its value depends 
on the diode transit time, according to Eqs. (7.13) and (7.16). If the forward excursion is arbitrarily 
restricted, for instance, to the point where the diode conductance rises to g max /5, the only way to 
reduce the conversion loss and noise figure with increased sinusoidal pumping is to use the reverse 
bias, resulting in a smaller pulsed current duty cycle. However, if the voltage swing is too far in 
the reverse direction, the loss will also increase rapidly due to the series resistance r s and junction 
capacitance Cj that lead to a residual diode conductance = r s (a>Cj) 2 , as shown in Figure 7.2. In 
this region, the diode is operated as a varactor with a quality factor Q v — l/(a> r s Cj). 




v 



IV 

FIGURE 7.2 Effect of diode parasitics on diode conductance. 



316 MIXERS AND MULTIPLIERS 



In the past, it was common to use varactor frequency converters to generate moderate to high 
levels of RF power. Their major advantage as the reactive devices over other types of the frequency 
converters was a little noise. The general relations between the real powers at all mixing frequencies, 
which were derived by Manley and Rowe, apply to any nonlinear reactance, including varactor as 
a nonlinear capacitor, driven by one or two incommensurable frequencies. Generally, if a nonlinear 
capacitor is excited by two frequencies f\ and/ 2 , the sideband frequencies of the form mfi + nf 2 are 
generated because of the varactor nonlinearity, resulting in the following two equations: 



mPmn 
mfi + nf 2 



E E 

m=Q n =—oo 

oo oo D 

W 'mil 

^ ^ mfi + nh 

n=0 m =—oo J J 



= 0 (7.17) 



(7.1* 



where P^ is the real power produced in the circuit at the frequency mfi + nf 2 [5,6]. These equations 
imply that the nonlinear capacitor is lossless and can be applied to parametric amplifiers, frequency 
upconverters and multipliers. 

In a frequency multiplier, there is only a single excitation signal with frequency f\ when/2 = 0, 
and n can be set to zero. Consequently, the summation over n in Eq. (7.17) can be omitted, all terms 
in Eq. (7.18) become zero, and Eq. (7.17) is simplified to 



X>m = 0 (7.19) 



where P m represents the power at the frequency mfi. Equation (7.19) states that applying the input 
signal with power P\ and frequency f\ to a lossless nonlinear capacitor must be converted to the 
output powers at the harmonics of f\ . In particular, for «;th-harmonic multiplier, the highest possible 
value of P m occurs when only Pi and P m are not zero, and the input power flowing to the capacitor 
is equal to the output power flowing from the capacitor, resulting in a multiplier efficiency of 100%. 
To achieve such an idealized condition, it is necessary to provide the varactor termination by pure 
reactances at all other harmonics. 

However, varactor mixers and multipliers are narrowband and very sensitive to slight mistuning that 
makes difficult their performance optimization. In addition, they can generate parasitic parametric 
oscillations due to capacitor nonlinearity depending on how strong varactor is pumped. Also, the 
small variation of pumping level results in a significant variation of the desired output power because 
the average varactor capacitance strongly depends on the voltage amplitude across the varactor 
junction. Thermal noise of the varactor series capacitance provides a fundamental limit on the noise 
performance obtainable from the frequency converters. The upper-sideband (USB) upconverter is 
generally stable, but its gain is limited, and for high-frequency signals no gain is possible. At the 
same time, the lower-sideband (LSB) upconverter is potentially unstable, and can have negative input 
and output resistances [6]. Therefore, the resistive diode mixers based on Schottky-barrier diodes are 
widely used in modern designs due to the circuit simplicity, performance stability and predictability, 
broadband design capability, and applicability up to millimeter-wave frequencies [7]. 

Figure 7.3(a) shows the microwave upconversion single-diode mixer (SDM) schematic where the 
signal from the LO flows through the directional coupler to the mixing diode connected in parallel, 
and the low-pass and bandpass filters are included into the IF and RF signal paths, respectively. In 
this case, conversion gain (loss) is the ratio of the output (RF) signal power to the input (IF) signal 
power. Isolation represents the amount of "leakage" or "feedthrough" between the mixing ports. 
Dynamic range is the amplitude range over which the mixer can operate without performance degra- 
dation. Harmonic intermodulation distortion results from the mixing of mixer-generated harmonics 
of the input signals having frequencies mfuo ± «/if, where m and n represent the harmonic order. 



SINGLE-DIODE MIXERS 317 






DPI 1 









BPF 



4 Jiv ,'ii> A Ji» 1 : 



Via- fir .1 



</.» 

FIGURE 7.3 Single-diode upconversion mixer and its spectrum. 



Cross-modulation distortion is the amount of modulation transferred from a modulated input signal 
to an unmodulated input signal when both signals are applied to the IF port. 

In upconversion mixers, the signal that is necessary to transfer to higher frequency band is applied 
to the IF port, the power level of the LO signal is increased, and then the high-side RF signal is filtered 
out, as shown in Figure 1 3(b). In this case, when both the LO and IF signals may have comparably 
high power levels, the spectrum of the output RF signal is rich with intermodulation products. For 
instance, if voltage applied to the mixing diode with exponential characteristic given by Eq. (7.12) 
represents the sum of two voltages, 



l) = Vq + V\ COS (xlyt + V 2 cos U>2t 



(7.20) 



where Vq is the dc bias voltage, then the currents corresponding to any intermodulation product in a 
diode mixer can be written as 



'id 



/sat exp 



Vo 



Vi 
nVj 



nVj 



cos (kcoit ± la>2t) 



(7.21) 



where hlV i/inVj)] a.ndI l [V2i'(nVj)] are the modified Bessel functions of the first kind of orders k and 
/ with argument Vil(nVj) and V / 2/(«V / t), respectively [8]. Here, the quantities k and / designate the 
current harmonic components produced in the diode mixer. As a result, the conversion loss of the SDM 
is sufficiently high, of about 10 dB. However, some improvement of 1-2 dB in the RF output power 
level can be achieved if to provide an optimum impedance for the second LO harmonic (2/ L0 ) at the 
RF output when its combining with a low-side RF frequency creates an intermodulation product of 
the secondary conversion f w — 2/ L0 — (/lo — /if) which is in phase with the correspondent product 
of the primary upconversion /rp =/lo +/if> similarly to that kind of process in a downconverted 
image-rejection diode mixer having optimum image-frequency impedance [9]. 



318 MIXERS AND MULTIPLIERS 



7.3 BALANCED DIODE MIXERS 

Balanced mixers are generally divided into two classes, called the single-balanced mixers and the 
double-balanced mixers, which combine two or more identical single-diode mixers. The advantage 
of balanced upconversion mixers over single-diode mixers includes both the rejection of spurious 
responses and intermodulation products and significantly better LO-to-RF, RF-to-IF, and LO-to-IF 
isolation. The level of rejection is dependent on the amplitude and phase balance of the baluns, 
providing the balanced drive and the matching between the diodes. 

7.3.1 Single-Balanced Mixers 

In a single-balanced mixer (SBM) circuit shown in Figure 7.4 where the two diodes as nonlin- 
ear elements are excited in phase by one signal and 180° out of phase by the other signal, no 
intermodulation outputs that include even-order harmonics of the out-of-phase signal exist [10]. 
In a general case when the transfer characteristic of the nonlinear element can be described as a 
polynomial 



the current in the upper nonlinear element can be calculated as 

oo oo oo 

(i = 22 ^ n c ° s a, 2 f + c ° s ^lO" = ^ n cos ° ^2'+ ^2 ^ m cosm mit 

11 =0 n=0 n=0 

/oo \ / oo \/°° \/°° \ 

+ 1 ^2 d p cos p a] 2 t 1 1 ^2 £q c ° sq °ht I — ( "52 ^ cos ' 0,21 ) I ^2 ^ s cosS 



(7.22) 



, P =i 



(7.23) 



By expanding each term into the Fourier series, Eq. (7.23) can be equivalently rewritten in 
frequency domain with harmonic representation as 



i'l = I ^2 Hi COS rb0 2 t I I 5^ ^ s COS S 



(7.24) 




■0 — r 



1 1 /, - 1 2 



FIGURE 7.4 Single-balanced diode mixer diagram. 



BALANCED DIODE MIXERS 319 



RF 



LO 



)J4 



OH 



ll 



FIGURE 7.5 Microstrip reflection-type branch-line diode mixer. 



Since both nonlinear elements are assumed to be fully identical, the current in the lower nonlinear 
element excited by the same input signal but with a 1 80° out-of-phase shift can be written as 



7 S cos sa>it 

,v=0 



H, cos r(m 2 t + 180°) 

r=0 

As a result, the total current (t that flows through the load is equal to 

oo 

^7/ r cos;-(o; 2 ? + 180 o ) 



(7.25) 



ij = /'i — i 2 = I J~]It cos s(Q\t I x | I H r cos rco 2 t 

M=0 / I V=0 



. (7.26) 



From Eq. (7.26) it follows that, for all r where r is an even integer, the current i' T becomes equal 
to zero. Therefore, all intermodulation products defined by frequencies of the form 



(Dim = | rb.vtoi ± rcool 



(7.27) 



exist for all s but only for odd r. 

The circuit for a microwave balanced upconversion mixer based on a branch-line hybrid is shown 
in Figure 7.5. Although not shown, generally any single-diode mixer (SDM) requires matching 
and bias circuits. Hybrid microwave combiners become practical for upconversion mixers when IF 
frequency is much smaller than both LO and RF frequencies, and both LO and RF frequencies differ 
insignificantly from each other to fit the hybrid operating bandwidth. For a fully symmetrical SBM 
circuit, the resulting current /rp corresponding to the USB RF frequency /rp = f LO + / IF can be 
written as 

*rf = -'rfi - z'rf2 = -Irf {cos [(oi LO t - —J + a> w t + 71 j 

+ cos (a) LO t + (Dtft + — ) J = 2/rf sin (&> L o + a>w) t. (7.28) 

where co\p — lirfw and culo — 27t/lo- 

Instead of a branch-line hybrid combiner, the rate-race ring hybrid combiner or coupled-line 3-dB 
coupler can be used [11,12]. The hybrid ring is more easily fabricated and controlled, but it is larger 
than the hybrid coupler and its operating frequency bandwidth is nowhere as good. Due to suppression 
of the even LO harmonics, the SBM provides less level of the intermodulation components in the 
transmitted RF signal spectrum than the SDM. When one mixing diode of an SBM fails, the circuit 
continues to operate, although the RF output power level drops by about 3 dB. 

To increase the operating frequency, a subharmonic mixer can be used that provides an ability 
to operate with halved LO frequency, thus avoiding signal leakage problems inherent with higher 
frequency LO sources. Figure 7.6(a) shows the simplified schematic of a subharmonic mixer with an 



320 MIXERS AND MULTIPLIERS 



RF o- 



II 



"^7 






J)LO 










W -3 





HI — |i' 



(a) 



Quasi-lumped 
open-circuited stub 



RF HPF 




Quasi-lumped 
short-circuited stub 



(b) 



FIGURE 7.6 Schematics of subharmonically pumped mixers. 



antiparallel mixing diode pair that presents an antisymmetric current-voltage characteristic [13]. In 
this case, the subharmonic mixer is equivalent to two SDMs in parallel, and each mixer is excited by 
a carrier LO signal at/Rp/2. For in-phase LO signals with reversed polarity of two parallel-connected 
diodes, as shown in Figure 7.6(a), the total current ij that flows through the RF load is equal to ij — 
h + i2 and, as it follows from Eq. (7.26) where the negative sign must be replaced by the positive 
one, this current becomes equal to zero for all r where r is an odd integer, including the fundamental 
component. Similar result at an RF load can also be achieved for 180° out-of-phase LO signals with 
the same polarity of two parallel-connected diodes. At microwave frequencies, the series resonant 
circuits tuned to /hp and/ L0 to isolate the LO and RF signal paths from each other can be replaced 
by the short-circuited and open-circuited microstrip stubs. In this case, the short-circuited stub at the 
LO port is quarter wavelength long at/ L0 , and becomes a half wavelength long at/Rp, thus providing 
a short circuit to the RF signal. On the other hand, the open-circuit stub at the RF output presents 
an open circuit to the RF but is a quarter wavelength long at/ LO , presenting a short circuit for LO 
signal. The frequency of the IF signal to be upconverted is normally far enough from/Rp to allow 
easy realization of an IF filter presenting an open-circuit output to the RF port. 

Figure 7.6(b) shows the circuit schematic of the subharmonically pumped (SHP) mixer MMIC 
[14]. The mixer includes an IF low-pass T-type lumped filter and RF high-pass 7-type quasi-lumped 



BALANCED DIODE MIXERS 321 



filter to isolate RF and IF ports. To isolate RF and LO ports, the quasi-lumped open-circuited and 
short-circuited stubs are used. The former stub provides a short circuit at/ LO with an open circuit 
at/RF, whereas the latter stub introduces a short circuit at/Rp, providing an open circuit at/u> The 
capacitances C\ and C 2 of the quasi-lumped open-circuited stub are defined as 



C, = 



Co 



1 

(UloZi tan 0i 



(7.29) 
(7.30) 



where Zj is the characteristic impedance and 9\ (0 < 9\ < 90°) is the electrical length of the stub 
at/ix). To effectively convert the IF signal to the RF, the quasi-lumped short-circuited stub should 
be grounded at IF frequency. In this case, the shunt capacitance C3 and electrical length # 3 can be 
calculated from 



~ 2a - ' / Z 3 \ 2 /l +3tan 2 0 2 N2 



tan6» 3 = — - ~ 1\ -3. (7.31) 

Z 2 2tan# 2 V V Z 2 / V 2tan<9 2 

1/1 Z 3 

C 3 = -tan 0 2 . (7.32) 

wlqZ 3 \tan0 3 Z 2 



where the electrical lengths 6 2 and 6* 3 , which are set at/ LO , must vary within the range of 0 < (6 2 , 
6 3 ) < 45° to short RF signal at the input of the short-circuited stub. 



7.3.2 Double-Balanced Mixers 

A double-balanced mixer (DBM) normally makes use of four diodes, which can be connected in 
different configurations, with both LO and RF signal paths being balanced. In this case, all ports of 
the mixer are inherently isolated from each other, and isolation is achieved by means of center-tapped 
transformers. The advantages of a double-balanced design over a single-balanced scheme are the 
increased linearity, improved suppression of spurious products, and inherent isolation between all 
ports. However, they require a higher level LO drive to pump the diodes and two additional baluns. 
The DBMs have higher conversion loss and lower limit in maximum frequency, but they provide 
broader bandwidth. 

Figure 7.7(a) shows the bridge double-balanced diode upconversion mixer schematic with two 
center-tapped transformers to provide isolation between three ports: LO, IF, and RF [15]. The LO 
voltage is assumed to be large enough to properly control on/off cycle of the diodes. The RF currents 
i\ and f' 3 , which are shown in Figure 7.7(b), flow through the diodes D\ and £> 3 , respectively, during 
the time when voltage dlo makes point a positive with respect to point b and voltage Dip makes point c 
positive with respect to point d. Diodes D\ and D 3 are turned on by v L0 , and current ;' LO flows around 
the loop a-c-b-a. In this case, diodes D 2 and D4 are turned off because they are reverse-biased. The RF 
currents ;'i and ; 3 add in the load resistance R L to produce the RF voltage with indicated polarity. Note 
that ;' L o does not flow through the IF path between the center points of the transformers because these 
points have the same potential at/ix> if the diodes and the LO transformer are perfectly balanced. 
The polarity of u IF in Figure 7.7(c) is the same as in Figure 7.7(b), but the polarity of v L0 is reversed, 
thus making point b positive with respect to point a. In this case, diodes D 2 and £> 4 are turned on, 
while diodes Di and D 3 are turned off. As a result, the RF currents i 2 and (4 flows into the RF load, 
producing the RF voltage of opposite polarity to that in Figure 7.7(b). The LO current flows through 
the loop involving diodes D 2 and £> 4 , but not in the RF transformer. This type of mixer can operate 
over a wide frequency range determined primarily by the design of transformers. If toroidal-cored 
transmission-line transformers are used, bandwidths of 1000:1 can be achieved. The balance of the 



322 MIXERS AND MULTIPLIERS 




(b) (c) 

FIGURE 7.7 Equivalent schematics of diode double-balanced mixer. 

mixer and isolation between ports is determined by the accuracy of the transformer windings and 
careful matching of the diode characteristics. 

Generally, the two most common types of DBMs are the ring mixer (RM) and the star mixer [7]. 
The RM is more relevant to low-frequency applications, in which wire-wound or transmission-line 
ferrite transformers can be used. The star mixer is used primarily in microwave applications, as it 
is more amenable to operate with microwave baluns. However, there is no significant difference in 
their performances. Figure 7.8(a) shows the classical diode RM circuit where the secondary windings 
of the transformers are connected to the nodes of the diode ring composed by four diodes. The LO 
signal alternately turns the top diode pair Di and D 2 and bottom diode pair £> 3 and £) 4 on and off 
in anti-phase, thus providing the corresponding closed loops for IF currents through the diodes and 
the grounded center tap of the LO transformer in opposite directions. In this case, if the diodes are 
identical, the points b and d represent virtual grounding to the RF signal, hence no RF voltage appears 
across the secondary winding of the LO transformer. Similarly, the points a and c are virtual ground 
to the LO signal, thus no LO voltage appears across the secondary winding of the RF transformer. If 
diode ring is fabricated on a single chip, the isolation between LO and RF ports in microwave mixers 
can be achieved from 30 to 40 dB . The IF signal can also be delivered through the center tap of the LO 
transformer rather than through the center tap of the RF transformer. The RM concept can be extended 
to include double-ring mixer topologies that have the added advantage of allowing the IF frequency 
response to overlap the RF and LO frequency bands, thus making easier the process of frequency 
selection [16]. In this case, to reduce the conversion loss and to extend the frequency upconversion 
to higher operating frequencies, the broadband active center-tapped balun can be implemented in 
microwave monolithic design. 

The upper-frequency limit of the DBMs using tapered baluns and low parasitic diode packages can 
also be extended if a balun structure is formed by two coupled transmission-line pairs, each having 
a quarter wavelength at the center bandwidth frequency, as shown in Figure 7.8(fo) [17]. Without 



BALANCED DIODE MIXERS 323 





any frequency compensation, this structure can exhibit greater than octave of bandwidth and can 
be realized in a variety of media, such as coaxial cable, microstrip, or coplanar waveguide. As a 
result, the upper-frequency limit of such a DBM occurs when the total length of the balun becomes 
360°. To minimize the overall mixer size for monolithic microwave applications above 10 GHz, the 
mixer architecture for frequency upconversion can include a miniature spiral balun for IF path and 
a 180° hybrid formed with an interdigitated Lange microstrip coupler with two (+45° and —45°) 
phase shifters. This allows us to split an LO incoming signal into two paths with equal amplitudes 
and 1 80° relative phase difference and to provide an output port for the RF extraction of upconverter 
application [18]. 

In a star DBM architecture shown in Figure 7.9, one terminal of each four diodes is connected to 
a common node used as the IF terminal [7,19]. Similarly to the RM, this mixer operates as a polarity- 
reversing switch. For the LO transformer T 2 , when the dotted sides of its secondary windings T 2 a 
and T2b are positive, diodes Di and Di are turned on, diodes D3 and D4 are turned off, and the dotted 



324 MIXERS AND MULTIPLIERS 




i o 

FIGURE 7.9 Schematic of diode double-balanced star mixer. 



sides of the secondary windings Ti a and of the RF transformer T\ are connected to the IF port. 
The RF port is thus connected to the IF port through the RF transformer T\ and the diodes. However, 
when the LO polarity reverses, diodes D\ and D 2 are turned off, diodes D 3 and Z? 4 are turned on, 
and then the undotted sides of both secondary windings and 7*ib are connected to the IF port. 
In this case, the RF port is again connected to the IF port, but with reversed polarity. As a result, 
the RF polarity is therefore always applied to the IF port, but its polarity is reversed periodically at 
the LO frequency. Because generally the operational principles of the star and RMs are the same, it 
should expect similar spurious-response properties, and the only difference may be defined by the 
convenience and complexity of the implementation in a particular design and technology. 

Single-sideband (SSB) mixer architectures are useful in discriminating and removing the LSB 
and USB generated during frequency upconversion, especially when sidebands are very close in 
frequency and attenuation of one of the sidebands cannot be achieved with filtering. Conventional 
SSB mixers generally include a Wilkinson divider to provide two LO signals, well balanced both in 
amplitude and phase, a 90° phase shifter to provide quadrature IF signals, and a quadrature hybrid 
coupler to combine the extracted RF signal at USB, as shown in Figure 7.10(a). Unlike the low-noise 
receiver downconverter, the transmitter upconverter operates with sufficiently high power levels (in 
limits of 5 to 10 dBm) at both IF and LO inputs, and therefore, the sidebands are rich with IF 
harmonics. However, since the even IF harmonics and their intermodulation components are well 
suppressed in each SBM, only the sidebands with the odd-order components such as/ix> ± (fw, 3/rp, 
5/rp, . . .) are delivered to the output quadrature coupler of the DBM. The carrier LO frequency is also 
suppressed in the SBMs, typically by about 20 dB, due to the corresponding isolation of their output 
hybrids. 

Consider the propagation paths of the main signal components &> LO ± a>rF from both SSB mixers 
through the output quadrature coupler-combiner in Figure 7. 10(a), by assuming that the phase delay 
in directions 1-3 and 2^1 is equal to 90° and there is no phase delay in diagonal directions. As a 



BALANCED DIODE MIXERS 325 



Single- 
balanced 
inker 




Single- 
bu kneed 
mixer 




O fat- f_ij ■ J..- 



(a) 



divider 



SI IP 
rTii\ur 



li, {^) 1 




JUL\0J' 



FIGURE 7.10 Block diagrams of single-sideband mixer-upconverters. 



result, for an USB RF signal at port 3, the phase delay from the top mixer is equal to 0 1-3 = (j) L0 + 
0if + 90°, whereas the phase delay from the bottom mixer is equal (j> 2 ,i — <Plo + (4>if + 90°) + 0°, 
which are in phase, and the USB RF signal /lo +/if flows to the RF load. At the same time, for a 
LSB RF signal, the in-phase condition is achieved at port 4, when the phase delay from the top mixer 
is equal to 0 1|4 = 0 L o — 4>w + 0°, whereas the phase delay from the bottom mixer is equal 0 2 ,4 = 
<Plo — (4>w + 90°) + 90°. As a result, the LSB RF signal /lo — /if is fully dissipated in the ballast 
resistor and does not appear in the RF load. 

The suppression level of an unwanted sideband strongly depends on a circuit symmetry level 
(equality of pumped LO signals, identity of voltage-ampere diode characteristics, equality of phase 
shifts to nominal values, etc.). For instance, if two RF signals having equal amplitudes and being 
orthogonal in phase flow from both top and bottom SSB mixer outputs, then their suppression level 
at the output of the non-ideal quadrature coupler-combiner can be evaluated according to 



1 + 2AAcosA0 + AA 2 
l-2AAcosA0 + AA 2 



(7.33) 



where AA is the ratio of amplitudes and A<p is the phase difference produced by the coupler. 
Consequently, in order to achieve the suppression level of an unwanted sideband of more than 20 
dB, it is necessary to provide AA and A<p to be better than ±0.2 dB and 10°, respectively. As an 
example, in a microwave SSB diode mixer operated as a frequency upconverter with IF power P w — 
5 dBm and LO power P L0 =10 dBm, an upconversion of the modulated IF to RF can occur with the 
conversion loss of 6 dB and LO suppression of 9 dB, relatively to the USB/rf =/lo +/n= [20]. 



326 MIXERS AND MULTIPLIERS 

TABLE 7.1 Intermodulation Components in RF Spectrum for Different Mixer Types. 



Harmonics LO 



Harmonics IF 


1 


2 


3 


4 


5 


6 


Mixer Type 


1 


+ 


+ 


+ 


+ 


+ 


+ 


SDM 




+ 




+ 




+ 




SBM 




+ 




+ 




+ 




UBM, KM 


2 


+ 


+ 


+ 


+ 


+ 


+ 


SDM 




+ 




+ 




+ 




SBM 
















DBM, RM 


3 


+ 


+ 


+ 


+ 


+ 


+ 


SDM 




+ 




+ 




+ 




SBM 




+ 




+ 




+ 




T"M"1 TV J" in li ft 

DBM, RM 


4 


+ 


+ 


+ 


+ 


+ 


+ 


SDM 




+ 




+ 




+ 




SBM 
















DBM, RM 


5 


+ 


+ 


+ 


+ 


+ 


+ 


SDM 




+ 




+ 




+ 




SBM 




+ 




+ 




+ 




DBM, RM 


6 


+ 


+ 


+ 


+ 


+ 


+ 


SDM 




+ 




+ 




+ 




SBM 
















DBM, RM 



To simplify filtering problem by mixing with a harmonic of a lower frequency LO signal, which is 
very important when IF frequency is not so high, and to increase an operating frequency bandwidth 
for RF signal towards millimeter waves, an SSB upconverter based on subharmonically pumped 
mixers with antiparallel pair of diodes can be used [14,21]. Figure 7. 10(£>) shows the block schematic 
of the wideband SSB SHP upconversion mixer consisting of two SHP mixers, an LO power divider 
with a phase shift of 45°, and an RF in-phase power combiner. In this case, the IF signals are fed into 
each SHP mixers having the phase difference of 90°, the LO signal is divided into two signals with 
the phase difference of 45°, and then the SHP mixers generate the USBs and LSBs of the upconverted 
signal 2/lo ±/if- Since upconverted USB signals 2/ L0 +/if are in phase with each other at the input 
of the Wilkinson combiner, hence the desired upconverted USB resulting signal appears at the RF 
output, while out-of-phase LSB resulting signal is canceled. 

To compare different diode mixer architectures, the presence or absence of various intermodulation 
component in an RF spectrum due to IF and LO harmonic components (from fundamental to sixth) is 
shown in Table 7.1, where the harmonic availability is marked by the "+" symbol. Thus, the spectrum 
of an SDM contains all possible harmonic components, unlike an SBM, where the even-order LO 
harmonics are suppressed. However, in a DBM or RM, the number of intermodulation components 
reduces even more, in total by four times, due to the corresponding suppression of both IF and LO 
even harmonics. 



7.4 TRANSISTOR MIXERS 

Transistor mixers can be built based on both bipolar transistor and FET device; however, GaAs FETs 
are commonly used at microwave frequencies due to their superior noise and gain performance. For a 
bipolar mixer, it was found that the frequency conversion, which is strongly dependent on the forward 
collector-emitter current gain of the device in a common base configuration in the low-frequency 
frequency range, primarily depends on the base resistance in the medium-frequency range, and is 



TRANSISTOR MIXERS 327 



mainly dependent on the base-emitter capacitance [22]. Due to its highly nonlinear characteristic, 
the FET device turns out to be a very efficient mixer, and its conversion transconductance exhibits a 
flat maximum as a function of the gate bias voltage. In this case, an optimum voltage range can be 
obtained where the ratio between mixing current and distortion product currents is most favorable. 
Generally, FET mixers exhibit smaller transconductance than mixer stages with bipolar transistors 
[23]. However, the FET input impedance is higher and thus does allow the use of better and more 
selective input resonance circuits. The more effective mixer performance can be achieved with lower 
gate capacitance and bulk resistances at higher frequencies where FET devices are preferred over 
bipolar transistors as having less intermodulation and cross-modulation distortion and their lower 
feedback capacitance provides better circuit stability. 

In a general case, if a device has the transfer characteristic 

/(f) = a 0 v(t) + ai v 2 (t) + a 2 v 3 (t) (7.34) 

then applying a two-tone input signal 

v(t) — Vi coscoit + Vi cosco 2 f (7.35) 

results in the following components of the output current provided correspondingly by first-order 
term 

agVi cos coit + a 0 V2 cos a>2t (7.36) 

second-order term 

— ' (V 2 + V 2 ) + y V 2 cos 2m x t V 2 2 cos 2&> 2 ? + fli Vi V 2 [cos (^ + a> 2 ) t + cos (&>j - a) 2 ) t] 

(7.37) 

and third-order term 

^3o2 F 3 + 3 _?l Vl vA cos co lt + {^Vl + ^V^c os co 2 t+ ^Vf cos 3eo lt 

a 2 -, 3a 2 , 

+ ^V 2 cos3m 2 t -\ — — Vj V 2 [cos(2a> 1 + a> 2 )t + cos (2a> l — co 2 )t] 

3ai t 

+ -~ Vi V 2 2 [cos (2(o 2 + o) l ) t + cos (2co 2 - &>i ) t ] (7.38) 

where the term a^v represents linear mixer action, reproducing the input signals at the output, the 
term a\v 2 gives rise to a dc component and second harmonics of the input signals as well as the 
product terms at frequencies/] if 2, and the term a 2 v 3 produces components at frequencies/!,/^, 
3/!, 3/ 2 , 2f x ±/ 2 , and 2/ 2 ±/j [15]. 

The basic block schematic of a single FET mixer, where an active device is treated as a three-port 
network, is shown in Figure 7.11(a), with each port generally intended to be connected to LO, IF, 
or RF signal path, respectively. However, conventional FET mixers usually employ a gate mixing 
to produce a desired frequency component when LO signal shares the same gate terminal with IF 
modulated signal. In this case, independent matching for each signal is impossible, and, to achieve a 
sufficiently high isolation between both LO and IF signals, it is necessary to use hybrid circuits such 
as Lange couplers, power dividers, or baluns that increase the mixer circuit complexity. 

Figure 7.1 1(b) shows the circuit schematic of a monolithic upconversion heterojunction FET mixer 
MMIC based on coplanar waveguides (CPW) as the transmission lines that can be used in microwave 



328 MIXERS AND MULTIPLIERS 



IFO-| H * 



II tiller 




FIGURE 7.11 Block diagram and circuit schematic of source-injection FET mixer. 



and millimeter- wave applications [24,25]. In this case, a source injection concept was introduced in 
which the IF signal with a frequency fjp is applied to the gate terminal through the IF filtering and 
matching network, the LO signal with a frequency /lo is applied to the source terminal through the 
LO filtering and matching network, and the resultant USB RF signal having a frequency /rp = /if + 
/lo is extracted from the drain terminal, after passing through the RF matching circuit and output 
filter. The RF output filter consists of two T-type networks connected in parallel for suppressing the 
LSB frequency /ip — /lo and LO frequency / L0 , but passing the USB frequency / IF +/lo- The LO 
matching network is essentially responsible to provide an unconditionally stable operation for the 
upconverter, both in presence and absence of an LO signal. In this structure, since the LO signal 
is directly applied to the source terminal, it has the largest effect on the device transconductance 
modulation and, consequently, on nonlinear mixing enhancement, which facilitates a low LO power 
operation. As a result, the upconverter can operate with an LO power level as low as — 16 dBm for IF 
signals in 1.5-2.5 GHz band, and LO signals in 20-23 GHz band. In this case, LO suppression at IF 
and RF ports is better than 20 dB, whereas IF suppression at RF port is better than 35 dB. 

Figure 7.12 shows the circuit schematic of a monolithic gate-injection HEMT mixer that shifts a 
signal in the 16 GHz band up to the V-band using a 48 GHz LO signal [26]. Such a mixer topology 
was chosen because it is simple in implementation and can achieve high conversion gain for low LO 
power levels. The network at the gate side of the mixer was designed to provide 50-Q matching at 
both IF (16 GHz) and LO (48 GHz) frequencies. However, since two signals are applied to the same 
port, the external diplexer is required. The gate bias network also includes a small (5 Q) resistor, 
which is needed to ensure unconditional stability in the circuit. The network at the drain side of 



DUAL-GATE FET MIXER 



329 



If 





Vi 



M 



iHi' 



cH H J - i r 



-O RF 



LO+ IF 



FIGURE 7.12 Circuit schematic of upconverting HEMT mixer. 



the device provides both the suppression of LO and IF signals and 50-Q. matching at RF (64 GHz) 
frequency. Obtaining a good suppression is important because it allows the HEMT device to remain in 
the saturation region over the entire LO cycle. This achieved by an open-circuit stub with an electrical 
length of 90° at 48 GHz. Measurements of the fabricated circuit demonstrated a peak conversion gain 
of 1 dB at 64.5 GHz for —1.7 dBm LO power, an LO suppression better than 30 dB, and an input 
third-order intercept point of — 1 .6 dBm. 



7.5 DUAL-GATE FET MIXER 

Dual-gate FET devices have a major advantage over their single-gate counterparts so that the LO and 
IF (or RF) signal paths can be connected to separate gates, thus providing a good LO-IF (or LO-RF) 
isolation because the capacitance between the gates is very low. Therefore, single-gate FET devices 
are mostly used in realizing balanced mixers since a balanced dual-gate FET mixer requires separate 
hybrids or baluns for splitting input signals that makes the mixer design too complicated. 

The dual-gate MESFET device and its decomposition into an equivalent cascode connection 
of two single-gate MESFET parts with RF grounded second gate is shown Figure 7.13(a) and 
Figure 7.13(A), respectively [27,28]. In this case, the first device is operated in a common source 
configuration, whereas the second device is operated in a common gate configuration, and the drain 
current I Al of the first device represents the input current of the second device. This means that the 
first transistor can operate as a current source for the second transistor, and the drain-source voltage 
Vfoi across the first transistor can be controlled by the gate bias voltage V gS 2 of the second transistor. 
The other advantage of the dual-gate FET device is that it can provide operation stability over wider 
frequency range due to much lower feedback capacitance between the second-device drain and the 
first-device gate. 

The main nonlinear operation regions of a dual-gate MESFET can be identified using its bidimen- 
sional transfer and output voltage-ampere transfer characteristics shown in Figure 7.13(c) [29]. Here, 
the vertical traces correspond to the different gate voltages V gS 2 of the FET2 for constant Vds — 5 V 
and must be divided between the channels of the two single-gate FETs because V& — Vdsi + Via- In 
this case, there are mainly three nonlinear regions of the dual-gate MESFET to operate as a mixer: 



• Region I where FET1 operates in a linear region of its output voltage-ampere characteristics 
and FET2 operates in a current saturation region with low V gs i and V gS 2 close to pinch-off 
voltages. 



330 MIXERS AND MULTIPLIERS 




5 4 3 2 1 0 J.' lU ,V 

(c) 

FIGURE 7.13 Dual-gate MESFET, its equivalent cascode representation, and voltage-ampere characteristics. 

• Region II where the FET1 operates in a linear region close to saturation and the FET2 operates 
in a current saturation for high V gs i and V gS 2 far from pinch-off voltages. 

• Region III where the FET1 operates in a saturation region and the FET2 operates in a linear 
region with high V gs i and V gs2 far from pinch-off voltages. 

Figure 7.14 shows the equivalent circuit of a dual-gate MESFET showing the elements responsible 
for nonlinear operation in different bias regions. It is obvious that in bias regions I and II, mixing 
takes place inside of FET1, while FET2 acts as a linear amplifier. In this case, if the mixing in region 
I is mainly provided at low drain current by the nonlinearity of the device transconductance g mi and 
output differential resistance i?d S i as the functions of V gs i, then, in region II, the nonlinear effect of 
the gate-source capacitance C gs i at high drain current becomes significant as well. Due to the low 
drain current, the mixer operating in region I requires smaller LO power and provides low-noise 
performance. However, if the dual-gate MESFET operates as a mixer in region III, the nonlinearities 
are located in FET2, and FET1 acts now as a linear amplifier. 

Figure 7.15 shows the block schematic of a microwave upconversion mixer based on a dual-gate 
GaAs MESFET operating at X-band with an IF input signal varying over 700 ± 250 MHz applied 
to the second gate [30]. Such an upconverter pumping by the LO signal at 7.4 GHz can offer the 



BALANCED TRANSISTOR MIXERS 331 



Sua 




FIGURE 7.14 Nonlinear equivalent circuit of dual-gate MESFET. 



conversion gains up to 15 dB, noise figure of 3.2 dB and saturated RF output power up to 12 dBm. 
For a particular case of the saturated output power of 9 dBm with the LO level set at 12 dBm, the 
third-order intermodulation product at the 1-dB compression point was —22 dBc. The isolation from 
the input LO port to the input IF port was measured to be 28 dB, whereas the isolation from the output 
RF port to either of the input ports was greater than 20 dB. 



7.6 BALANCED TRANSISTOR MIXERS 
7.6.1 Single-Balanced Mixers 

A pair of single-gate MESFET mixers can be combined into a single-balanced MESFET mixer using 
different hybrid combiners or simply parallel connection at the output. In this case, the properties 
of balanced MESFET mixers with regard to intermodulation products and isolation of input and 
output ports are essentially the same as in balanced diode mixers. Figure 7.16 shows the typical block 
schematic of a balanced MESFET mixer where the LO signals being 1 80° out of phase are mixed with 
IF signals fed from a 180° hybrid [31]. As a result, the output upconverted signals are combined at 
the RF port as being in phase, while the LO signals are canceled as being out of phase. In the practical 
layout of such a 20-GHz balanced upconversion mixer, a 180° phase difference between the LO 
signals can be obtained through the slotline F -junction formed on the other side of substrate. The 



IF 
input 



LO 
input 



O 



Filtering 

and 
Matching 



Filtering 

and 
Matching 



G2 



Gl 



Filtering 

and 
My idling 



RF 

output. 



FIGURE 7.15 Dual-gate MESFET upconverter configuration. 



332 MIXERS AND MULTIPLIERS 



LO 
input 



IF 
input 



Filtering 

and 
Male hi rig 



180" 



Filtering 

and 
Matching 



RF 
output 



Filtering 

and 
Match i tig 



FIGURE 7.16 Balanced MESFET upconverter configuration. 



combiner can consist of a coupled microstrip lines with a connecting gold ribbon whose position is 
properly adjusted to optimize the LO reflection phase. As a result, the LO level measured at the output 
RF port was lower by 12 dB than the upconverted output power of 15.9 dBm without external filter. 

Since the input MESFET impedance for IF frequency is high, the gate voltage swing of the IF signal 
is considerably large. Consequently, a simplified analysis shows that the MESFET transconductance 
gmO) in time domain can be approximated by its saturation value g m o at positive half cycles of the IF 
signals and zero at negative half cycles. A Fourier series expansion of g m (t) results in 

/l 2 \ 
gm(0 = £mo I - + — smcospt H 1 (7.39) 

where a>w — 27t/if,/if is the IF frequency. When the LO voltage of a radian frequency <«lo = 2b/lo 
and an amplitude Ulo is applied, the drain current ( d (?) can be expressed as 

kit) = g m (t)V h0 sina) LO f 

= gmoVLo | ^sincuLO? - ^ [cos (m LO + co w ) t - cos (co LO - co w ) t] H J (7.40) 

which means that the RF output signal is lower by the factor of \ln (or —9.9 dB) corresponding to 
both low-side upconversion at co LO — &)jp and high-side upconversion at oj lo + o>if than the maximum 
RF output signal when the MESFET is used in a power amplifier application. 

For millimeter- wave applications, a concept of subharmonic balanced mixer with the LO frequency 
about half the RF output frequency can be used. Figure 7.17 shows the simplified circuit schematic of 
a 60-GHz balanced CMOS upconversion mixer where a miniature transformer-coupler is introduced 
to provide two LO input signals with 90° phase difference [32]. The IF signal feeds through the 
low-pass filter and injects to transistor Mi . By using the property of a common source transistor, the 
IF signals at the drain and gate of Mi have 180° phase difference at low frequency. Therefore, the IF 
signals flowing to the mixing transistors M 3 and M 4 are out of phase. The drains of M 3 and M 4 are 
connected together to cancel the fundamental IF and second harmonic LO. As a result, the measured 
LO-RF and 2LO-RF isolations for RF frequency from 58 to 66 GHz were better than 40 dB. 

Figure 7.18 shows the circuit schematic of a balanced Ku-bwA upconverter using HEMT tech- 
nology for low-noise monolithic application [33]. The mixer consists of an HEMT pair with the 
common source and common gate transistor configurations. For a common gate HEMT, the LO 



BALANCED TRANSISTOR MIXERS 



V M 
Q 



J 



o 



Z 



-|| ORF 



A/4 



11 i 



I 

I 



FIGURE 7.17 Schematic of balanced CMOS upconversion mixer MMIC. 



■H 



IF O- 




Hi- 



Hi' 



HI — t — II — 0 



Hi' 



FIGURE 7.18 Circuit schematic of balanced HEMT upconversion mixer MMIC. 



334 MIXERS AND MULTIPLIERS 



signal is injected between the gate and source ports. In both configurations, the time-varying device 
transconductance g m (t) is the dominant contributor to frequency conversion, and the effect of other 
nonlinearities is minimal. By properly selecting the gate width of the common source transistor, it is 
possible to have the same gain in both devices. As a result, the IF and LO signals will be canceled at 
the output of the parallel-connected transistors due to the phase shift (ideally 180°) in the common 
source transistor. To combine the LO and IF signals, two active circuits based on the single-stage 
amplifying stages were used, as well as a high-pass filter at the RF output was included to improve 
isolation. As a result, a conversion gain over 4.2 dB at RF frequency of 14 GHz (IF = 1.885 GHz) 
was achieved with 3 dBm of LO power and isolation between any ports over 27 dB. 



7.6.2 Double-Balanced Mixers 

The double-balanced transistor mixers have essentially the same characteristics as the double-balanced 
diode mixers in terms of isolation, spurious responses, or bandwidth but substantially higher con- 
version gain and they do not require hybrids or baluns that make them very attractive for compact 
monolithic implementation. The DBM in the form of a pair of the transistorized differential amplifiers 
was originally invented by Howard Jones to use as a dual output synchronous detector [34]. However, 
Alberto Bilotti was the first who described and analyzed the fully balanced arrangement of the three 
differential pairs shown in Figure 7.19 with regard to its different monolithic applications such as 
balanced product mixer (downconverter or upconverter) and suppressed carrier modulator [35]. In 
this case, a DBM requires low-level operation at both inputs, whereas suppressed carrier modula- 
tion is obtained by low-level modulating signal V\(t) and high-level carrier signal v 2 (t). In the latter 
case, carrier high-level operation provides a modulated output independent from the carrier level and 
should be preferred unless the harmonic spectral response is acceptable. Note that the amount of 
carrier suppression is a function of the matching accuracy of the differential pairs. 




FIGURE 7.19 Monolithic double-balanced mixer configuration. 



BALANCED TRANSISTOR MIXERS 335 





For an upconversion balanced mixer, both the input signal V[(t) and the input signal v 2 (t) can 
respectively be considered either an LO signal or an IF signal to be converted into the output RF 
signal v ou i(t). Being implemented with a GaAs HBT technology, such a double balanced upconverter 
when V[(t) — Vjp(t) and v 2 (t) — v L0 (t) can provide the conversion gain greater than 20 dB (with buffer 
amplifiers) up to an RF output frequency of 5.5 GHz [36]. At the same time, a 20-GHz AlGaAs/GaAs 
HBT frequency upconverter can exhibit the conversion gain of 5 dB with the RF-LO isolation of 
23 dB for RF output up to 8.5 GHz when v x {t) = v LO (t) and v 2 (t) = v w (t) [37]. 

Since generally the differential-pair DBM requires external baluns for the LO and IF input ports 
resulting in its bulky and expensive implementation, one way to solve this problem is to use a 
lumped equivalence of the hybrid baluns. GaAs material has the semi-insulating substrate with low 
conductivity, and therefore, a sufficiently high quality factor of the lumped-element inductors can be 
achieved. Figure 7.20 shows the circuit schematic of a fully integrated double-balanced GalnP/GaAs 
HBT upconversion mixer, consisting of a mixer core based on two differential pairs (Q 5 to Q s ) with 
a 180° lumped rat-race hybrid for the balanced LO signal, an active IF balun (<2i to Q4), and an RF 
output LC current combiner with an emitter-follower output buffer (Qg and gio) [38]. The rat-race 
hybrid employs the 7r-type LC low-pass network and the T-type LC high-pass network to replace the 
quarter-wavelength and three-quarter-wavelength transmission-line sections, respectively. The output 
LC current combiner formed by two inductors of 1.1 nH and one capacitor of 0.43 pF is necessary to 
perform the differential-to-single (or balanced-to-unbalanced) conversion at the RF output port. As a 
result, the conversion gain of 1 dB when /if = 300 MHz,/lo = 4.9 GHz, and/Rp = 5.2 GHz can be 
achieved, with the measured LO-RF isolation of 38 dB. The single-to-differential conversion for the 
LO signal can be provided by an active balun in the form of the differential HBT pair reducing the 
overall die size but consuming extra power with limited dynamic range and conversion gain. In this 
case, in order to increase the conversion gain, the output differential amplifier is used providing also 
the differential-to-single conversion of the output RF signal [39]. 

Figure 7.21 shows the simplified schematic of a monolithic double-balanced MESFET mixer that 
upconverts an input 100 to 500 MHz IF signals to an output 0.6 to 1.75 GHz RF signals with a 



336 MIXERS AND MULTIPLIERS 



O 



UK 

output 



IF 

input 



O- 



-o 



IF 
input 



LO 
input 



O- 



1.0 
input 



Q, 



o. 



T 



FIGURE 7.21 Simplified schematic of double-balanced MESFET mixer. 



conversion gain of 8 dB [40]. The circuit consists of two mixers stages that are excited by IF and LO 
signals that are 0° and 180° out of phase with respect to one another. The outputs of the two mixers 
are then combined and amplified to produce an RF output, with the conversion gain depending on the 
LO level. Mixing action is accomplished by alternating amplification of the two 180° out-of-phase 
IF signals at the LO frequency. When the LO transistor is turned on, ideally an ac ground is created 
across it, which makes an effective common-source amplifier for the IF signal. Conversely, when the 
LO transistor is turned off, no amplification of the IF signal occurs at the output. Therefore, the LO 
signal alternately turns each side of the mixer into an IF amplifier, thus creating an output RF signal 
that is a combination of the two signals. To achieve the proper mixing action, the dc components of 
the LO and IF transistors must be identical to ensure in the quiescent state that the drain-to-source 
voltages of the devices Q3 and Q$ are zero, thereby preventing any LO feedthrough in the absence of 
an IF signal. 

For low supply voltage and low power consumption requirements, RF CMOS technology can be 
used in the high microwave range in phased-array radars, wireless local networks, local multipoint 
distribution services (LMDS), and other industrial-scientific-medical (ISM) band applications. Al- 
though passive mixers have the inherent advantage of better frequency and linearity response, they 
have conversion loss and also require a high LO drive level. A typical circuit schematic of the double- 
balanced CMOS mixer based on a pair of differential amplifiers is shown in Figure 7.22(a) where 
the gate and source degeneration inductors are used to match LO and IF inputs and RF output to 
standard 50 Q. [41]. Having source degeneration inductors also helps to improve the linearity at some 



BALANCED TRANSISTOR MIXERS 




338 MIXERS AND MULTIPLIERS 



expense of conversion gain. To improve conversion gain of the upconversion mixer that is intended 
to operate in a superheterodyne 60-GHz transmitter, the transmission-line stubs in the output circuit 
and between each differential pair and current-source transistors can be used [42]. 

However, even lower available LO drive can be achieved in a dual-gate DBM, the simplified 
circuit schematic of which is shown in Figure 7.22(A) [41]. In a dual-gate mixer, a finite LO signal 
must be presented at the drains of lower devices (Mi to M 4 ), which causes the devices to operate 
between linear and saturation regions, thus resulting in the modulated transconductance of these 
devices. This is because the lower devices are operated in linear region during most part of the LO 
cycle, unlike a conventional double-differential mixer where switching action of the LO transistor 
pair occurs between the cutoff and saturation regions. As a result, the conversion gain of a dual-gate 
mixer is slightly lower, but its linearity performance is moderately better. The mixer can achieve a 
conversion gain of 1.7 dB and an output 1-dB compression point of —3.4 dBm at a supply voltage of 
1.2 V with a current consumption of 9 mA when/ IF = 2,3 GHz and/m; = 23 GHz. 



7.7 FREQUENCY MULTIPLIERS 

The frequency multipliers historically were the very important part of the vacuum-tube transmitters 
to achieve a high-frequency transmission of a stable modulated signal. In this case, a high-frequency 
transmitter chain was comprised of a crystal-controlled vacuum-tube circuit, oscillating on the crystal 
frequency, and a non-oscillating power amplifier or several power amplifiers, the output resonant 
circuit of which is tuned to a small integral multiple of the crystal frequency [43,44]. It was found 
that, for any given excitation voltage, there is a fairly critical adjustment of negative grid voltage 
corresponding to Class C mode for maximum efficiency and another critical adjustment for maximum 
output harmonic power. As the order of multiplication is increased, these adjustments become more 
critical and the points of the maximum harmonic efficiency and power more nearly coincide. However, 
both higher efficiency and greater output power is obtained for higher values of excitation voltage. 

For a piecewise-linear approximation of the device transfer characteristic shown in Figure 7.23 
with cosine driving voltage, an effect of the duty cycle (or conduction angle) of the pulsed output 
current on the operation mode of a frequency multiplier can be readily understood based on the ratio 
of the Hth-harmonic current amplitude /„ and maximum amplitude 7 max of the output current, which 
can written as 

oin = Y a - (7.41) 

'max 

where maximum current amplitude 7 max as a function of a half-conduction angle 9 is equal to 

/max = /U-COS0) (7.42) 



where / is the pulsed current amplitude, and / = 7 max when 9 = 90°, which corresponds to 50% 
duty cycle. The amplitude 7 n of ;ith current harmonic is defined as / n = y n (6)I, where y n {9) is the 
^-harmonic current coefficient defined as 



/„(#) = 



1 



sin(« — 1)8 sin(« + 1)( 
n (n — 1) n (n + 1) 



(7.43) 



As a result, 



" 1 — COS0 



(7.44) 



FREQUENCY MULTIPLIERS 339 



At'ih 'rail 















\ 




A 


0 




- tin u 

n-w 


it 2i: 



col 



fur 



FIGURE 7.23 Piecewise-linear approximation and pulsed output current waveform. 

which shows that, for each harmonic component, there is a particular angle of current flow, for which 
a n is a maximum, and hence where the maximum harmonic output is obtainable. In this case, the 
maximum value of a n (6) is achieved when 6 — 1207», where n is the harmonic order. The best 
angle of current flow is almost exactly inversely proportional to the order of the harmonic and differs 
slightly for the piecewise-linear and square-law cases [45]. 

The possibility of a frequency multiplication in the transistor amplifiers can be explained on the 
basis of two physical principles. The first principle is based on the transistor operation with a certain 
conduction angle that results in the harmonic generation at the device output. In this case, it is enough to 
employ idler circuit at the collector of the bipolar transistor tuned to the selected harmonic component, 
as shown in Figure 7.24(a), where the tank idler circuit tuned to the nth harmonic is used to provide 
low impedance conditions at the collector for the rest harmonic components including fundamental. 
Such a circuit configuration of the transistor frequency multiplier is usually used for low orders of 
frequency multiplication and high-power levels [46]. The second multiplication principle is based on 
the nonlinearity of the device collector-junction capacitance that is used in the varactor parametric 
multipliers [6]. In this case, due to the load-network configuration with series idler harmonically- 
tuned circuits with finite loaded quality factors, as shown in Figure 7.24(A), the collector current 
also flows through the feedback collector abrupt-junction capacitance, thus resulting in a frequency 
multiplication due to parametric effect. The parametric frequency multiplication is a very effective 
approach when it is necessary to achieve the output frequency exceeding the transistor transition 
frequency / T by 2-3 times. In addition, for lower power transistors and higher order multiplication, 
the idler circuit can be placed at the base of the transistor that exhibits the abrupt change of base- 
spreading resistance as the input voltage increases that may result in an additional improvement of 
the multiplier conversion gain [47]. 

With introduction of GaAs MESFET devices operated as frequency multipliers, the high conver- 
sion gain and good isolation between the input and output ports due to significantly lower feedback 
gate-drain capacitance can be achieved. Besides, the MESFET frequency multiplier provides high 
conversion efficiency and circuit stability, requires low driving power, and the lack of an idler circuit 
at the input makes it reasonable broadband [48]. Generally, the major nonlinearities in the MESFET 
causing harmonic generation are the nonlinear gate-source and gate-drain capacitances, the drain 
current clipping due to the gate-source Schottky diode, and the nonlinear transconductance and out- 
put differential resistance due to the drain current source operation in the pinch-off and saturation 
regions. However, the largest contributor to the MESFET frequency doubler is the drain-current 
clipping effect when the device is biased near close to 0 V, just below the forward conduction point 



340 MIXERS AND MULTIPLIERS 




(7...I 



FIGURE 7.24 Circuit schematics of bipolar frequency multipliers. 



of the gate-source junction, resulting in a half-rectified waveform and a maximum conversion gain 
for the MESFET frequency doubler [49]. This suggests that the device is principally a resistive dou- 
bler and therefore harmonic conversion gain is expected to fall as (l/«) 2 . In a dual-gate MESFET, 
the frequency multiplication is provided primarily by the combining effect of the input diode and 
drain current source nonlinearities [50], In this case, the first gate transconductance is periodically 
modulated by an input signal and the amplified signal swing at the second gate is modulated in turn 
by its nonlinearity, thereby generating harmonics. Then, the resultant harmonics are further amplified 
and extracted from the drain. As a result, the dual-gate MESFET multiplier offers the advantages of 
better efficiency and easier gain control over its single-gate counterpart. 

Figure 7.25 shows the simplified schematic of a reflector-type MESFET frequency doubler where 
the fundamental-frequency matching circuit and second-harmonic reflector are placed at the input, 
and the second-harmonic matching circuit and fundamental-frequency reflector are placed at the 



FREQUENCY MULTIPLIERS 341 



Matching circuit Reflector Reflector Matching circuit 

for in,, for 20)0 for u>„ for 




FIGURE 7.25 Schematic of reflector-type MESFET frequency doublet'. 



output [51,52]. The open-circuit stub with an electrical length of 45° is located inside the input 
reflector to provide the short-circuit condition for the second harmonic, while the open-circuit stub 
with an electrical length of 90° and series stub with an electrical length of 180° represent the output 
reflector to provide the short-circuit condition for the fundamental at the device drain. As a result, a 
conversion gain of 6 dB for the MESFET frequency doubler was achieved at an output frequency of 
24 GHz. Using the separate fundamental-frequency rejection network at the device drain when it can 
be adjusted independently from the frequency-selective load network can improve significantly the 
conversion gain for the 1 1.5 to 34.5 GHz HEMT frequency tripler [53]. The fundamental-frequency 
termination at the device output is especially useful for a millimeter-wave HEMT multiplier when 
the frequency ranges from 180 to 220 GHz for a doubler and from 130 to 150 GHz for a tripler with 
acceptable conversion gain can be achieved [54]. 

To provide high spectral purity and broadband operation, the balanced frequency-doubler config- 
urations are preferred because they provide efficient rejection of the fundamental and odd-harmonic 
components. In this case, the bandwidth of the doubler is mainly limited by the phase and amplitude 
imbalance of the hybrid that increases with frequency. To overcome this limitation, it is necessary to 
apply broadband MMIC approaches for combining scheme or hybrid implementation. Figure 7.26(a) 
shows the miniaturized broadband balanced frequency doubler MMIC that consists of a common-gate 
and common-source MESFETs connected in parallel electrically and in series by electrode configura- 
tion such as source-gate-drain-gate-source [55]. Here, the gate of a common-gate device is grounded 
through a capacitor, and the source of a common-gate device and the gate of a common-source device 
are connected in phase at the input port through the phase shifter and dc-blocking capacitor, respec- 
tively. The phase shifter compensates for a phase error between the device outputs, and the output 
matching circuit is required for load matching at the second harmonic. The doubler operates near the 
gate-source pinch-off voltage, and the amplitudes of the output signals from both MESFETs are made 
equal by adjusting each gate bias. Due to different device configurations, their fundamental compo- 
nents are 180° out of phase, whereas the second-harmonic components are in phase, thus resulting in a 
cancelation of the fundamental frequency signals and an in-phase combining of the second-harmonic 
signals. The conversion gain from —8 to —10 dB was achieved with fundamental-signal suppression 
better than 17 dB from 6 to 16 GHz. 

Figure 7.26(b) shows the basic configuration of a balanced frequency doubler consisting of the two 
identical MESFETs and a 1 80° hybrid divider [56]. Since a practical rat- race hybrid was characterized 
by amplitude imbalance of about 1.1 dB and phase imbalance of about 2° from 28.8 to 39.4 GHz, the 
bias compensation approach when each transistor is biased differently to compensate for the hybrid 
asymmetry was applied. As a result, the fundamental-frequency rejection achieved was better than 
35 dB over second-harmonic output frequency bandwidth of 31.5 to 37.5 GHz. A planar Marchand 
balun is very effective to divide the incoming signal into two equal parts with 1 80° phase difference 
in millimeter- wave applications to provide a conversion gain of —10 dB for a balanced HEMT 
doubler covering the frequency range from 150 to 220 GHz [57], To achieve frequency doubling over 
an octave bandwidth, good conversion gain and fundamental-frequency rejection, the microwave 
MESFET frequency doubler can be built based on a balanced structure with Lange couplers followed 



342 



MIXERS AND MULTIPLIERS 



Phase 
shifter 

r t = K u 




(c) 

FIGURE 7.26 Schematics of balanced MESFET frequency doublers. 



by a broadband balanced amplifier, as shown in Figure 7.26(c) [58]. In this case, the input and output 
Lange couplers in a frequency doubler are connected symmetrically, rather than in complementary 
symmetry, as for a conventional amplifier, thereby creating a frequency doubled waveform with 
rejected fundamental signal to be then amplified. The bandwidth of the doubler is controlled by the 
bandwidth over which the Lange couplers can maintain a 180° phase-difference path. An average 



FREQUENCY MULTIPLIERS 



343 



Short 
for 3u>i 



Short 
for 2 tub 



Input 



Short 
for 2d>, 



Open 

for *Ud 



1 



Output 

1 31* 



(a) 



Rctlcctoi Rctlectoi 

for 3o>, for 2(Dr, 



Reflector 
for 2t% 



Input 



— K 



" ■ 1 1 ~ 



1 



45° 



Reflector 

for dju 



^ ( lutpul 

3ca» 



FIGURE 7.27 Circuits schematics of reflector-type HEMT frequency tripler. 



conversion gain of — 3 dB with fundamental-frequency rejection greater than 12 dB was achieved in 
the 8-16 GHz output band. 

Figure 7.27(a) shows the simplified circuit schematic of a lumped HEMT tripler where the 
architecture of the input and output circuits is chosen to exploit the fact that large amounts of unwanted 
harmonic signals exist at both the transistor input and output [59]. For a tripler, the schematic includes 
resonant circuits for 2&>o and 3a>o at the gate and for coo and 2a>o at the drain to reflect these unwanted 
harmonics back into the transistor gate and drain, causing nonlinear interactions that can be utilized 
to maximize the resultant 3a>o output signal. Optimization of the idealized input and output circuits 
shows that maximum conversion gain is provided when both input resonant circuits represent a short 
circuit for 2w Q and Txdq, whereas the output resonant circuits should represent an open circuit for 
co 0 and a short circuit for 2&> 0 , respectively. Besides, inserting the offset transmission lines between 
the harmonic loads and the transistor from both sides can further improve the tripler performance by 
varying the phase of the harmonic signals. In practical microwave implementation, the short circuit 
elements (input 2a> 0 and 3&>o and output 2a> 0 ) are realized using corresponding quarter-wavelength 
open-circuit stubs, as shown in Figure 7.21(b). The open circuit for a>o was implemented with a 
parallel combination of chip capacitor and wire-loop inductor, allowing reflection of the fundamental- 
frequency signal without affecting 3co 0 output. As a result, for such a reflector-type HEMT tripler, 
the measured conversion gain greater than 3.5 dB for the input fundamental-frequency signal at 2.94 
GHz was achieved. 

Figure 7.28 shows the simplified circuit schematic of a balanced MESFET tripler with quadrature 
input and output couplers [60]. In this case, the input Ku-band signal is amplified in the first stage 
and feeds the tripler MESFETs with 90° out of phase. At the output ports of the second stage, the 
fundamental-frequency signal remains with 90° out of phase, whereas the second and third harmonic 
signals are 1 80° and 270° out of phase, respectively. When combined in the output quadrature coupler, 
only third-harmonic <2-band signals are in phase. The fundamental-frequency components are 180° 
out of phase and cancel each other at the output. The fundamental suppression of 40 dB over frequency 
range from 42 to 5 1 GHz was achieved using Lange couplers. 



344 MIXERS AND MULTIPLIERS 




/t2u„)- l«0° 
^3o>o) = 270« 



FIGURE 7.28 Schematic of balanced MESFET frequency tripler. 



REFERENCES 



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Sept. 2008. 



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22. J. Zawels, "The Transistor as a Mixer," Proc. IRE, vol. 42, pp. 542-548, Mar. 1954. 

23. J. S. Vogel, "Nonlinear Distortion and Mixing Processes in Field-Effect Transistors," Proc. IEEE., vol. 55, 
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24. M. Madihian, L. Desclos, K. Maruhashi, K. Onda, and M. Kuzuhara, "A Monolithic AlGaAs/InGaAs Up- 
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25. M. Madihian, L. Desclos, K. Maruhashi, K. Onda, and M. Kuzuhara, "60-GHz Monolithic Down- and Up- 
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26. A. Orzati, F. Robin, H. Meier, H. Benedikter, and W. Bachtold, "A V-Band Up-Converting InP HEMT Active 
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202-204, June 2003. 

27. S. Asai, F. Murai, and H. Kodera, "GaAs Dual-Gate Schottky-Barrier FET's for Microwave Frequencies," 
IEEE Trans. Electron Devices, vol. ED-22, pp. 897-904, Oct. 1975. 

28. C. A. Liechti, "Performance of Dual-Gate GaAs MESFET's as Gain-Controlled Low-Noise Amplifiers and 
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29. C. Tsironis, R. Meierer, and R. Stahlmann, "Dual-Gate MESFET Mixers," IEEE Trans. Microwave Theory 
Tech., vol. MTT-32, pp. 248-255, Mar. 1984. 

30. W. S. Tsai, S. F. Paik, and B. S. Hewitt, "An X-Band Dual-Gate FET Up-Converter," 1979 IEEE MTT-S Int. 
Microwave Symp. Dig., pp. 495^197. 

31. T. Hirota and H. Ogawa, "A Novel K-Band Balanced FET Up-Converter," IEEE Trans. Microwave Theory 
Tech., vol. MTT-32, pp. 679-683, July 1984. 

32. P.-S. Wu, C.-H. Wang, C.-S. Lin, K.-Y. Lin, and H. Wang, "A Compact 60 GHz Integrated Up-Converter Using 
Miniature Transformer Couplers with 5 dB Conversion Gain," IEEE Microwave and Wireless Components 
Lett., vol. 18, pp. 641-643, Sept. 2008. 

33. M. L. de la Fuente, J. Portilla, J. P. Pascual, and E. Artal, "Low-Noise Ku-Band MMIC Balanced P-HEMT 
Upconverter," IEEE J. Solid-State Circuits, vol. SC-34, pp. 259-263, Feb. 1999. 

34. H. E. Jones, "Dual Output Synchronous Detector Utilizing Transistorized Differential Amplifiers," U.S. Patent 
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35. A. Bilotti, "Applications of a Monolithic Analog Multiplier," IEEE J. Solid-State Circuits, vol. SC-3, pp. 
373-380, Dec. 1968. 

36. A. Y. Umeda, C. T. Matsuno, A. K. Oki, G. S. Dow, K. W. Kobayashi, D. K. Umemoto, and M. E. Kim, 
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37. K. Osafune and Y. Yamauchi, "20-GHz 5-dB-Gain Analog Multipliers with AlGaAs/GaAs HBT's," IEEE 
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38. C. Meng, T.-H. Wu, and M. Lin, "Compact 5.2-GHz GalnP/GaAs HBT Gilbert Upconverter Using Lumped 
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688-690, Oct. 2005. 

39. K. W. Kobayashi, L. T. Tran, S. Bui, A. K. Oki, D. C. Streit, and M. Rosen, "InAlAs/InGaAs HBT X-Band 
Double-Balanced Upconverter," IEEE J. Solid-State Circuits, vol. SC-29, pp. 1238-1243, Oct. 1994. 

40. M. West and M. Kumar, "Monolithic Integrated Blanking Up-Converter," IEEE Trans. Microwave Theory 
Tech., vol. MTT-38, pp. 1227-1231, Sept. 1990. 

41 . A. Verma, K. K. O, and J. Lin, "A Low-Power Up-Conversion CMOS Mixer for 22-29-GHz Ultra- Wideband 
Applications," IEEE Trans. Microwave Theory Tech., vol. MTT-54, pp. 3295-3300, Aug. 2006. 



346 MIXERS AND MULTIPLIERS 



42. A. Valdes-Garcia, S. Reynolds, and J.-O. Plouchart, "60 GHz Transmitter Circuits in 65 nm CMOS," Proc. 
2008 IEEE RF1C Symp., pp. 641-644. 

43. R. S. Ohl, "High Frequency Oscillation Generator," U.S. Patent 1,756,131, Apr. 1930 (filed Aug. 1925). 

44. R. M. Page, "An Investigation of the Phenomena of Frequency Multiplication as Used in Tube Transmitters," 
Proc. IRE, vol. 17, pp. 1649-1655, Sept. 1929. 

45. F. E. Terman and J. H. Ferns, "The Calculation of Class C Amplifier and Harmonic Generator Performance 
of Screen-Grid and Similar Tubes," Proc. IRE, vol. 22, pp. 359-373, Mar. 1934. 

46. H. C. Lee and R. Minton, "Designing Transistor Multipliers," Microwaves, vol. 4, pp. 18-28, Nov. 1965. 

47. G. D. O'clock and R. J. Dauphinee, "High-Gain Transistor Frequency Multiplier," Proc. IEEE, vol. 58, pp. 
1363-1365, Sept. 1970. 

48. J. J. Pan, "Wideband MESFET Microwave Frequency Multiplier," 1978 IEEE MTT-S Int. Microwave Symp. 
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49. A. Gopinath and J. B. Rankin, "Single-Gate MESFET Frequency Doublers," IEEE Trans. Microwave Theory 
Tech., vol. MTT-30, pp. 869-875, June 1982. 

50. P. T. Chen, C.-T. Li, and P. H. Wang, "Dual-Gate GaAs FET as a Frequency Multiplier at Ku-Band," 1978 
IEEE MTT-S Int. Microwave Symp. Dig., pp. 309-3 1 1 . 

51. C. Rauscher, "High-Frequency Doubler Operation of GaAs Field-Effect Transistors," IEEE Trans. Microwave 
Theory Tech., vol. MTT-31, pp. 462^73, June 1983. 

52. Y. Iyama, A. Iida, T. Takagi, and S. Urasaki, "Second-Harmonic Reflector Type High-Gain FET Frequency 
Doubler Operating in K-Band," 1989 IEEE MTT-S Int. Microwave Symp. Dig., pp. 1291-1294. 

53. G. Zhang, R. D. Pollard, and C. M. Snowden, "A Novel Technique for HEMT Tripler Design," 1996 IEEE 
MTT-S Int. Microwave Symp. Dig., pp. 663-666. 

54. Y. Campos-Roca, C. Schworer, A. Leuther, and M. Seelmann-Eggebert, "G-Band Metamorphic HEMT- 
Based Frequency Multipliers," IEEE Trans. Microwave Theory Tech., vol. MTT-54, pp. 2983-2992, July 
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55. T. Hiraoka, T. Tokumitsu, and M. Akaike, "A Miniaturized Broad-Band MMIC Frequency Doubler," IEEE 
Trans. Microwave Theory Tech., vol. MTT-38, pp. 1932-1937, Dec. 1990. 

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1985. 

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60. H. Fudem and E. C. Niehenke, "Novel Millimeter Wave Active MMIC Triplers," 1998 IEEE MTT-S Int. 
Microwave Symp. Dig., pp. 387-390. 




Oscillators 



This chapter presents the principles of oscillator design, including start-up and steady-state 
operation conditions, noise and stability of oscillations, basic oscillator configurations using lumped 
and transmission-line elements, and simplified equation-based oscillator analyses and optimum de- 
sign techniques. An immittance design approach is introduced and applied to the series and parallel 
feedback oscillators, including circuit design and simulation aspects. Voltage-controlled oscillators 
(VCOs) and their varactor tuning range and linearity for different oscillator configurations are dis- 
cussed. Finally, the basic circuits and operation principles of crystal and dielectric resonator oscillators 
are given. 

8.1 OSCILLATOR OPERATION PRINCIPLES 
8.1.1 Steady-State Operation Mode 

A simple feedback oscillator model is shown in Figure 8. 1 (a) where an oscillator circuit is decomposed 
into a forward nonlinear network and a feedback linear network, both of which are two-port networks. 
Figure 8.1(b) shows an example of a transformer-coupled metal-oxide-semiconductor field-effect 
transistor (MOSFET) oscillator without bias circuit to illustrate common features of the feedback 
oscillator. Because an oscillator is an autonomous circuit, electronic noise in the active device or 
power supply turn-on transient leads to the self-excitation of the oscillations. This provides the initial 
oscillation build-up. As the oscillation amplitude grows, the active device displays larger nonlinearity 
and then limits the amplitude increase. 

In a steady-state operation mode, the following complex equation also known as the Barkhausen 
criterion can be written for the parallel feedback oscillator: 



where A = I ant IV\ n is the forward transfer function and B — Vi n // out is the feedback transfer function. 
This equation means that the oscillator complex loop gain is equal to unity [1,2]. The feedback 
transfer function can be represented as 



where K — ViJV out is the voltage feedback coefficient and Z = V out /I oat is the oscillator resonant 
circuit impedance. Presenting each of these complex quantities in the form of A — Aexp(j<f> A ), K — 
Kexp(j4>^), and Z = Zexp(j<t>z), the following equations for magnitudes and phases directly result 
fromEq. (8.1): 



T ( Vin, co) = A(V in , jco) B (jco) = 1 



(8.1) 



B(jto) = K (jco) Z (jco) 



(8.2) 



A (Vm, to) K (co) Z (co) = 1 



(8.3) 



<pA + 4>K + <t>Z = 



0, 2;r, . . . . 



(8.4) 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



347 



348 OSCILLATORS 





A{V in , o ) 




V 






out 




























(a) (b) 
FIGURE 8.1 Schematics of (a) parallel feedback oscillator and (b) transformer-coupled MOSFET oscillator. 



Equation (8.3), which is called the amplitude balance condition, means that the oscillator loop 
gain is equal to unity in the steady-state stationary operation mode. In this equation, it is assumed that 
two quantities, K and Z, depend on frequency only. Consequently, the amplitude balance condition is 
satisfied only under the appropriate value of input voltage amplitude V in . To define the value of this 
amplitude, let us rewrite Eq. (8.3) in the form 

A(V ln ,a»)= 1 (8.5) 
K (co) Z (a>) 

In Figure 8.2, the amplitude dependence A(V in ) and feedback straight line l/KZ are plotted. The 
intersection point of these dependencies determines the steady-state oscillation amplitude V ; °. 

Equation (8.4), which is called the phase balance condition, means that the sum of all oscillator 
loop phase shifts must be equal to zero or 2jtn, where n — 1,2, ... . This equation defines the value of 
the oscillation frequency / osc . In the simple case when (p K — 0 (transistor input admittance is equal to 
zero and feedback magnitude K depends only on a mutual-coupling coefficient M between primary 
and secondary inductances), and the active device does not produce the phase shift, that is <f>A — 0, 
then 4>z — 0, and the oscillation frequency / osc is equal to parallel resonant circuit frequency /o = 
(OqI(2tt), where a> 0 — 1/*JLC. If 4> A 0, the oscillation frequency / osc will differ from/o in order to 
fully compensate for the available phase shift <p z according to 

^z^-tan- 1 2<2 ) = -0a (8-6) 

where Q — il(coCR) is a quality factor of the oscillator resonant circuit, Aw = 2jz(f ox —fo) [2]. 




FIGURE 8.2 Graphic balance amplitude condition. 



OSCILLATOR OPERATION PRINCIPLES 349 





6W2 CWl 
Am, 



Q\>Qi 



Aw> 

(«) (b) 

FIGURE 8.3 Deviation of oscillation frequency / osc from resonant frequency /q. 




Equation (8.6) can be rewritten in the form 



An) 
o> 0 



tan0 A 
~2Q~ 



(8.7) 



which determines the deviation of the oscillation frequency / osc from resonant frequency /o as a 
function of the phase of the forward transfer function of the active device and oscillator quality factor. 

From graphical representation of Eq. (8.6) shown in Figure 8.3(a), it follows that the oscillation 
frequency /osc is smaller than the resonant frequency fo- Furthermore, the lower the quality factor 
Q the smaller oscillation frequency / osc , as shown in Figure 8.3(fo), where the deviation Aa> 2 from 
radian resonant frequency m 0 becomes greater for the case of Q 2 < Q\ . 



8.1.2 Start-Up Conditions 

The start-up conditions can be illustrated on the plane of the transfer function ! out =f(v in ), as shown 
in Figure 8.4. Let us choose the bias voltage V g ' corresponding to the maximum small-signal transfer 
function when di 0 uildv m — 0. Such a condition is adequate to the maximum forward transfer function 
A(Vjn = 0) shown in Figure 8.2. According to Figure 8.2, the amplitude V in of the oscillations grows 
monotonically, whereas the transfer function A decreases. The amplitude of the output current i mt 
is trying to reach its maximum value corresponding to the stable steady-state operation mode when 
y in — y£. Such a behavior of the transfer function A means that the start-up oscillation conditions 
can be defined as 

A(co) K((d) Z(o>) > 1 (8.8) 

0A + <fe + <fc=O,27T,.... (8.9) 

Let the bias voltage K g " be chosen close to the device threshold voltage, as shown in Figure 7.4. 
Such a bias condition corresponds to the small initial value of transfer function A(V in — 0), which 
is not enough to establish the stable steady-state stationary oscillations because A < \IKZ. This 
is demonstrated by curve // in Figure 8.5(a). Therefore, the oscillation system with such a bias 
condition requires an impulse to initiate self-oscillations, as it cannot oscillate by itself. The result 
of this impulse should be output current amplitude resulting in a condition of A > l/KZ. Hence, the 
system has hard operating conditions and the process of the oscillation establishment is called the 



350 OSCILLATORS 




hard build-up of self-oscillations. This means that oscillations of finite amplitude are established 
suddenly under some external influence. 

Figure 8.4 shows the process of establishment of the stable self-oscillations as a result of the 
nonlinearity of the transfer function of the active device operating in pinch-off and active regions. 
This means that the start-up oscillation conditions correspond to a Class A operation mode of the 
active device having maximum or close to maximum value of the small-signal transconductance. 
The steady-state oscillation conditions are established when the active device operates in Class AB 
mode characterized by certain conduction angle whose particular value depends on the initial bias 
conditions. 

Let us assume a high value of the resonant circuit quality factor when the input cosinusoidal 
voltage 

Win = Vi n COSft>? (8.10) 



CI 




FIGURE 8.5 Balanced amplitude conditions for different biasing points. 



OSCILLATOR OPERATION PRINCIPLES 351 



is applied to the active device, representing an idealized nonlinear voltage-controlled current source. 
Then, the output current i' ou , contains the harmonic components and, being an even function, can be 
written as 

'out — Io + h cos cot + I 2 cos 2cot + / 3 cos 3cot + • ■ ■ . (8.11) 

For the transfer function ( out = /(i>i n ), the fundamental component I\ can be obtained from 
Eqs. (8.10) and (8.11) using a Fourier formula according to 



/(V in cosa>f)cosc<;rd(6jf). (8.12) 



Consequently, dividing the output fundamental amplitude Ii by the input voltage amplitude Via 
provides an analytical expression to calculate the fundamentally averaged (average) value of the 
forward transfer function A(V in ) in a quasilinear approximation. The dependence of the output 
fundamental current amplitude I\ on the input voltage amplitude V m , expressed generally as 

/i = /i(V to ) (8.13) 

can be called the amplitude characteristic of the oscillator. Figure 8.5 shows the dependencies of 
(a) the amplitude characteristic and (b) average transfer function as functions of the input voltage 
amplitude Vm for different operation modes where curves / and curves // correspond to the soft and 
hard start-up conditions for self-oscillation establishment, respectively. 

Consider the influence of the harmonic components on the process of the establishment of the self- 
oscillations for finite value of the quality factor Q. Then, the input voltage according to Figure 8. 1(b) 
can be written as 

Vm = Vini cos cot + V m2 cos (2cot - — ^ + Vi„ 3 cos (?,a>t — ^ H (8.14) 

where the resonant circuit is tuned to the fundamental and has the capacitive reactance for the second- 
order and higher order harmonics. If we provide a second-order polynomial approximation of the 
nonlinear transfer function of the active device in the form 

'out = + aiu in + a 2 vf n (8.15) 

and to confine our attention to the first two components in Eq. (8.14), the fundamental component of 
the output current can be obtained by 

i[ — Ii costof + a 2 V in i Vi n 2 cos ((at — — ^ (8.16) 

where /i = a{V iTl i. 

Equation (8. 16) can be rewritten in the form 

i[ = I[ cos {cot + (f> A ) (8.17) 

where 



! + V .n2 (8-18) 



352 OSCILLATORS 



and 



tan(/) A = — 



«2 V,„2 



(8.19) 



Thus, the second component in Eq. (8.16) results in the variation of the output current fundamental 
amplitude when I[ ^ I { and the appearance of a negative value of the phase <p A of the device transfer 
function that means the phase shift between the fundamental voltage and fundamental current. In this 
case, an increase in the harmonic content of the output voltage spectrum (smaller g-factor) causes a 
decrease in the frequency of self-oscillations, as follows from Figure 8.3(fe). 

Physically, the influence of the harmonic content on the frequency variation can be explained 
as follows: if the oscillations are purely sinusoidal, the energy distribution in both arms of the 
resonant LC-circuit is equal; when the harmonics appear, the currents corresponding to them flow 
mainly through the capacitive arm, and therefore they increase the electrostatic energy of this arm 
in comparison with the inductive arm; and in order to keep the energy equal in both arms, the 
fundamental frequency must slightly diminish itself in respect to the frequency given by the tank 
circuit only [3]. 

In Figure 8.6(a), for the example of the transformer-coupled MOSFET oscillator shown in Fig- 
ure 8.1(£>), the dependencies /i(V in ) for a soft start-up condition with different values of mutual 
coupling coefficient M (M t < M 2 < M 3 < M 4 ) are shown [1,2], As M increases from zero to M 2 , 
the only stable equilibrium corresponds to the static operation mode at the point V m — 0. When 
M > M 2 , from two potentially existing equilibrium conditions the dynamic conditions at the points 
A 3 and A 4 are stable. To check the stable equilibrium at the point A 3 with amplitude V in = V;° and 
unstable at the Vm = 0, assume an oscillation rise of the small input amplitude due to some 
internal or external effect. This would cause an appearance of the output current with the amplitude 
/{, which is determined according to the oscillator amplitude characteristic /i(Vi„). At that time this 
current makes conditional the input voltage with amplitude V^. As a result, the oscillation amplitude 
that has arisen accidentally increases up to the equilibrium value of V° at the point A 3 . With the 
growth of M, the output current amplitude 1\ changes smoothly, as shown in Figure 8.6(£>). When 
M decreases, the amplitude /j changes in accordance with Ii(M) curve and under M — M 2 the 
oscillations disappear. 

In Figure 8.7(a), the dependencies /i(V in ) for hard start-up condition with different values of 
mutual coupling coefficient M (Mi < M 2 < M 3 < M 4 ) are shown [1,2], At the point M — M 3 




0 



0 



in 



in 



in 



Mi My Mi M 



Ut) 



(h) 



FIGURE 8.6 Start-up conditions with different values of mutual-coupling coefficient M. 



OSCILLATOR CONFIGURATIONS AND HISTORICAL ASPECT 353 




FIGURE 8.7 Hard start-up condition versus mutual-coupling coefficient M. 



both curves intersect at three points, corresponding to three stationary modes: 0 is an equilibrium 
condition, B 3 and A3 are the dynamic modes with amplitudes and V£, respectively. From the 
process of changing of the output fundamental current amplitude Ii with initial voltage deviations 
from and VjJJ, it follows that the stable conditions correspond to the points 0 and A3, whereas the 
condition in the point B 3 is unstable. 

Let us define the dependence of output fundamental current amplitude I[ versus feedback coef- 
ficient M. As M increases from zero up to M 4 when the straight line corresponding to M 4 and the 
curve /i(V in ) are tangents in the origin, the only stable equilibrium corresponds to the static operation 
mode at the point V- m — 0 when the small fluctuations can not produce an oscillation arise. At the 
point M — M 4 , the above-mentioned operation mode becomes unstable and small oscillations grow 
up to large amplitude value corresponding to the point A 4 . The subsequent increase of M leads to 
the amplitude change along the curve h(M), as shown in Figure 8.7(fo). If, then, to decrease M down 
to M2, the collapse of the oscillation can occur only at the point Ai because the dynamic operations 
modes corresponding to the points A 3 andA 4 are stable. This fact results from the qualitative process 
examination of Figure 8.7(a), where the points A 2 and B 3 are unstable at the same time. Hence, the 
hysteresis region takes place between the points A2 and A 4 in limits of Mi < M < M 4 . Thus, the hard 
build-up of the self-oscillation is characterized by spasmodic development of the oscillation with 
large amplitude under smooth increase of the feedback coefficient M and spasmodic collapse of the 
oscillation under smooth decrease of M. 



8.2 OSCILLATOR CONFIGURATIONS AND HISTORICAL ASPECT 

The first oscillator configurations using vacuum tubes were based on the electromagnetic coupling 
between the output and input circuits. By providing a close enough coupling with the output, sufficient 
energy is supplied to the input circuit to keep the continuous self-sustaining oscillations. In this case, 
the oscillation frequency is approximately equal to the resonant frequency of the parallel LC-circuit 
if the coupling and active device parasitics are small enough. 

There are two basic configurations of a transformer-coupled oscillator. The oscillator configuration 
with a parallel resonant circuit at the input electromagnetically coupled with the output, as shown in 
Figure 8.8(a), is called the Armstrong oscillator in honor of Edwin H. Armstrong who first developed 
and described the condition of obtaining the self-sustaining oscillations using such a configuration [4] . 
Approximately at the same time, Alexander Meissner in Germany described the transformer-coupled 



354 OSCILLATORS 




oscillator with a parallel resonant circuit at the output electromagnetically coupled with the input, as 
shown in Figure 8.8(b), which is called the Meissner oscillator [5]. High-purity stable oscillations 
can be obtained by a fully balanced version of Meissner oscillator with a three-turn link providing 
an excellent isolation of the resonant circuit when very little energy is taken from the resonator to 
provide the voltages to drive the input and sustain the oscillations [6]. 

To satisfy a phase balance condition in the transformer-coupled oscillator, the transformer should 
provide a phase shift of 180°. If the primary and secondary windings of the transformer have the 
same direction of wind, it is necessary to connect the secondary winding in the opposite direction 
relatively the primary one, that is, to connect the end of the secondary winding where the voltage is 
in-phase with the anode (drain or collector) voltage to the ground. The coupling coefficient of the 
transformer is chosen to provide a soft start-up oscillation mode. 

The schematic of the Hartley oscillator is very close to the schematic of the Meissner oscillator [7]. 
The difference is that the tank inductor L having an additional output replaces the transformer, as shown 
in Figure 8.9(a). The inductance ratio determines the feedback coefficient. The Hartley oscillator can 
be represented by a well-known inductive three-point configuration shown in Figure 8.9(b), where the 
feedback element from the output to the input is the capacitor C. The inductors L { and L 2 represent 
the output and input circuits, respectively. 

Unlike the Hartley oscillator with electromagnetic coupling, the schematic of a Colpitts oscillator is 
based on electrostatic coupling using the capacitive divider, the capacitance ratio of which determines 
the feedback coefficient, as shown in Figure 8.10(a) [8]. The Colpitts oscillator can be represented 
by a well-known capacitive three-point configuration shown in Figure 8.10(b), where the feedback 
element from the output to the input is the inductor L [9]. The capacitors Ci and C2 represent the 
output and input circuits, respectively. 




OSCILLATOR CONFIGURATIONS AND HISTORICAL ASPECT 



355 




Depending on which electrode of the active device is grounded, we can distinguish three basic 
configurations representing a Colpitts family of oscillators based on the MOSFET devices, as shown 
in Figure 8.1 1. The Colpitts oscillator with grounded (common) source is shown in Figure 8.1 1(a), the 
Colpitts oscillator with grounded (common) drain is shown in Figure 8.11(b), and the Colpitts oscil- 
lators with grounded (common) gate is shown in Figure 8.1 l(c, d). The connection of the appropriate 
electrode of the tube (or transistor) to the ground have important effects upon the following: 

• The manner in which the dc power is fed to the oscillator and the corresponding loading effects 
on the resonant circuit. 

• The manner in which the output power is fed to the external load. 

• The distribution of the stray elements to the ground plane, which has important effects, partic- 
ularly at microwaves. 



Calculating the output impedance Z ou( for a common source oscillator shown in Figure 8.11(a) 
and a common drain oscillator shown in Figure 8.11(£>), which are identical, shows that grounding 
of any terminal of the oscillator circuit does not change its electrical performance, provided there 
are no changes in the connection of the feedback elements and load to the active device. Therefore, 
the schematic with a common gate shown in Figure 8.11(c) is characterized by the same electrical 
behavior as the oscillator circuits with common source and common drain. However, in such a 
configuration there is no connection of any load terminal to the ground that makes its practical 
implementation more complicated. Figure 8.11(0") shows a common gate oscillator schematic with 
the load connected in parallel to the inductance L, that is, between the drain and gate terminals. 
However, the other load connection, such as this with one grounded port, results in different electrical 
properties of the oscillator compared with common source or common drain configuration. 

The Gouriet-Clapp oscillator is a variation of the Colpitts oscillator with a tank inductor replaced 
by the series combination of an inductor L and a variable capacitor C v , as shown in Figure 8.12(a). 
In this case, frequency stability is improved because the reactance of such a series circuit varies 
more rapidly with frequency than that of a single inductor. However, the possibility to improve the 
oscillator stability by connecting a capacitor in series to one or each circuit inductor was first found 
based on a Hartley type of the oscillator [ 1 0] . In modem practical oscillator design, the Gouriet-Clapp 
oscillator configuration is called the Clapp oscillator, although G. Gouriet and J. Clapp had devel- 
oped it independently [11]. A parallel counterpart of the Clapp oscillator shown in Figure 8.12(b) 
was described by E. Seiler and, therefore, is called the Seller oscillator [11]. Such an oscillator 
configuration is useful for wideband frequency tuning when the capacitance C v in the parallel circuit 
is variable. 



356 OSCILLATORS 




OSCILLATOR CONFIGURATIONS AND HISTORICAL ASPECT 357 



7T 




(a) 



L 




(b) 



FIGURE 8.13 Schematics of Vackar oscillator. 



The Vackar oscillator is the modified Clapp oscillator with additional variable capacitor C v , 
whose simplified equivalent circuit representation is shown in Figure 8.13(a) [11,12]. It combines 
the features of the circuits with the series and parallel arrangements and is useful for very wideband 
frequency tuning. When C v = 0, the schematic of Vackar oscillator becomes similar to the schematic 
of the Clapp oscillator shown in Figure 8.12(a). The configuration of the Vackar oscillator shown 
in Figure 8.13(b) demonstrates the main difference compared with the Clapp schematic. Here, the 
capacitance C V 2 represents the phase-varying capacitance providing an additional phase shift. With 
this circuit, it is possible to utilize the maximum value of the oscillator quality factor over the complete 
tuning range, and the circuit has substantially constant output amplitude. Both oscillator schematics 
are equivalent, since the n -representation of the capacitances C v , C x , and Ci are replaced by the 
equivalent T-representation of the capacitances C v i, C V 2, and C V 3. 

The transformer-coupled oscillator based on a differential amplifier is shown in Figure 8.14(a), 
where the coupling coefficient is chosen to provide a stable soft start-up condition. The simple 
oscillator configuration using a differential transistor pair is shown in Figure 8.14(i). Since the 
voltage at the gate of the transistor connected to the parallel resonant circuit is in-phase to the voltage 
at the drain of the transistor with grounded gate, the feedback in such a differential-pair oscillator is 
positive. 

The push-pull connection of the transistors in power amplifiers is usually used to increase the 
resulting output power, simplify the output matching with load, and improve spectral performance 
by suppressing even-order harmonics. The same concept can also be applied to the oscillator design. 
The push-pull oscillator circuit shown in Figure 8.15(a) is based on the two single-ended Meissner 
oscillators, where the transistors are turned on and turned off alternately. Since the voltage at the gate 
of the one transistor is in-phase to the voltage at the drain of the other transistor, there is no need to 
invert phase using the secondary winding. Another push-pull oscillator configuration based on two 
single-ended Seiler oscillators is shown in Figure 8.15(£>), where the positive feedback is formed by 
the capacitive divider based on the capacitors C\ and C%. 



358 OSCILLATORS 




FIGURE 8.14 Schematics of differential-pair oscillators. 



8.3 SELF-BIAS CONDITION 

To provide more effective operation mode of the oscillator, a self-bias resistor is usually included in 
the oscillator circuit. Figure 8.16 illustrates the circuit principle of self-biasing by two examples. In 
a common source transformer-coupled oscillator shown in Figure 8.16(a), the self-bias resistor R s is 
shunted by the capacitor C s to minimize the RF signal losses. In a common gate oscillator whose 
schematic is shown in Figure 8.16(b), the RF choke is connected in series with the self-bias resistor 
R s for the same purpose. 

Let us consider the principle of the self-bias operation. The start-up conditions are satisfied under 
the initial gate-source bias corresponding to the large value of the small-signal transconductance when 
V s = 0. As the oscillation amplitude grows, the dc bias gate-source voltage V gs = V g — V s decreases 
due to dc voltage drop across the self-bias resistor with the dc collector current increase (K s > 0), as 
shown in Figure 8 . 1 7(a) . The decrease of the gate-source bias voltage leads to an appropriate decrease 
of the large-signal transconductance and to the high-efficiency steady-state operation mode with the 
gate-source fundamental voltage amplitude V- ln . As a result, the self-bias condition combines the soft 
self-excitation of the oscillation with high efficiency of the Class AB, Class B, or Class C operation 
mode under hard self-excitation of the oscillation with a certain value of the conduction angle. 



SELF-BIAS CONDITION 359 





(a) 
L 





!l 1 


f 

= c 3 



0>) 

FIGURE 8.15 Schematics of push-pull oscillators. 

Using a piecewise-linear approximation of the device transfer characteristic, the dc drain current 
/o as a function of the input gate voltage amplitude V m can be determined by means of the conduction 
angle as 

I 0 = gmVinYoW) (8.20) 

where g m is the device small-signal transconductance, yo(8) is the dc current coefficient, and 



COS0 



V'in 



(8.21) 



where 20 is the conduction angle and V p is the device pinch-off voltage. 

Substituting Eq. (8.21) into Eq. (8.20) allows us to obtain the relationship between gate-source 



V„ = V„ 



Io cos (9 
8m Vo(9)' 



(8.22) 



360 OSCILLATORS 




(a) 




ibi 

FIGURE 8.16 MOSFET oscillators with self-bias resistors. 



On the other hand, a similar relationship can be written by 

V gs = V g - I 0 R S . (8.23) 

Figure 8.17(A) shows the graphical solution of a system of these two equations, where Eq. (8.22) 
with the conduction angle of 28 — 360° is plotted by curve 1, Eq. (8.22) with the conduction angle of 
28 < 180° is plotted by curve 2, and Eq. (8.23) is plotted by curve 3. The intersection of curve 3 with 
curve 1 at the point A corresponds to the start-up oscillation mode, whereas the point B corresponds 
to the steady-state oscillation mode with constant amplitude. During the oscillation build-up, the dc 
drain current increases by the value of A/rj. 

Consequently, during the process of self-oscillations build-up, the two separate processes occur 
simultaneously: the increase of the oscillation amplitude across the resonant circuit and the decrease 
of the bias gate-source voltage due to the presence of the self-bias resistor R s . Generally, both these 
processes are interdependent. The velocity of the first process is defined by the time constant of the 
resonant circuit given by 

Q 

T = — (8.24) 

where Q — coqCRl is the oscillator quality factor at the resonant frequency a>o. The velocity of the 
second process is defined by the time constant of the self-bias circuit as 



7s = R S C S . 



(8.25) 



SELF-BIAS CONDITION 361 




If r s 3> T, the self-oscillations grow very rapidly, corresponding to the soft start-up condition for 
the bias voltage of V g — V s — V g (V s — 0). The intersection of the oscillator amplitude characteristic 
h(Vm) with ^ s — 0 and feedback line provides the voltage amplitude across the resonant circuit 
corresponding to point 1 shown in Figure 8.18(a). The dc drain current grows with the same velocity, 
resulting in the slow growth of the self-bias voltage V s due to the slow transient response of the self- 
bias circuit. This process contributes to the gradual transition to the points 2 and then 3, characterized 




(a) 

(<-) 

FIGURE 8.18 Self-pulse modulation and self-modulation phenomena. 



362 OSCILLATORS 



by different types of the oscillator amplitude characteristic and smaller voltage amplitudes, with 
further collapse of the self-oscillations. The dc drain current becomes zero, resulting in the onset 
of the discharging process of the self-bias circuit. When the discharging process is finished, the 
process of the self-oscillations build-up will start once again. Thus, such a self-oscillation process 
is accompanied by the self-pulse modulation, as shown in Figure 8.18(6). For a smaller difference 
between T s and T, the process of the oscillation amplitude decrease may not reach the collapse 
point when the oscillation amplitude starts to grow up. As a result, the self-oscillations have an 
amplitude modulation in the steady-state operation mode. This process is called the self-modulation. 
To eliminate the self-modulation effect, it is necessary to choose the condition when T s < T. 



8.4 PARALLEL FEEDBACK OSCILLATOR 

The determination of the start-up and steady-state oscillation conditions is very often based upon a 
loop or nodal analysis of the circuit. However, the oscillator analysis using matrix techniques brings 
out the similarities between several types of the oscillators and results in one group of equations, which 
can be used to analyze different oscillator configurations [13]. In this case, a two-port network can 
represent both the active device and feedback element. Depending on the oscillator configuration with 
parallel or series feedback, using of admittance Y- or impedance Z-parameters can be respectively 
applied. 

A generalized equivalent circuit of the parallel feedback oscillator is shown in Figure 8.19(a), 
where Y\, Y 2 , and Y 3 are the feedback elements. To calculate the steady-state stationary operation 
mode, it is convenient to add the matrix of the parallel feedback element Y 2 and the matrix of the 
active device F-parameters according to 



m + py = 



Yn + Y 2 
Y 2l - Y 2 



Y l2 - Y 2 
Y 22 + Y 2 



(8.26) 



-O — * 



(a) 




s O » 



-* O s 



(b) 



FIGURE 8.19 Equivalent circuit of parallel feedback oscillator. 



PARALLEL FEEDBACK OSCILLATOR 363 



In this case, a system of two equations for the two-port network input current / in , output current 
I out , input voltage V in , and output voltage V out can be written as 



Im = (Yn + Y 2 ) V m + (Y n - Y 2 ) V out 
hux = (Y 2l - Y 2 ) V w + (Y 22 + Y 2 ) V out . 



(8.27) 
(8.28) 



Since, for the oscillator shown in Figure 8.19(a), the boundary conditions are obtained in the form 



Im = ~YiV m 
I out — — i3 Vout 



(8.29) 
(8.30) 



the following matrix equation can be written as 



+ Y, + Y 2 Y l2 - Y 2 
Y 2 \ — Y 2 Y 22 + 12 + ^3 



V oll 



= 0. 



(8.31) 



Thus, the steady-state stationary condition can be derived by equating determinant to zero, 



I'll + Yi + Y 2 Y 12 - Y 2 
Y 2 \ — Y 2 Y 22 + Y 2 + Y$ 



= 0. 



After some simplification, Eq. (8.32) can be rewritten by 

(Y l2 -Y 2 )(Y 2l -Y 2 ) 



Y 22 + Y 2 + Y 3 



Y n + Yi + Y 2 



(8.32) 



(8.33) 



As a result, the steady-state oscillation condition for the parallel feedback oscillator represented 
as a one-port negative conductance oscillator can be written as 



Y aat + Y L = 0 



(8.34) 



where Y 3 — Y L , and 



^22 + Y 2 



(Y l2 -Y 2 )(Y 2l -Y 2 ) 
Yn + Y! + Y 2 



(8.35) 



Consequently, the separate equations for real and imaginary parts of the admittances for a negative 
conductance oscillator, which are similar to Eqs. (8.3) and (8.4) for magnitude and phase of the loop 
gain of a parallel feedback oscillator corresponding to the steady-state oscillation process, can be 
obtained as 



ReF 0Ut + ReK L = 0 
Imy out + ImK L = 0. 



(8.36) 
(8.37) 



Similarly, the start-up conditions for the parallel feedback oscillator given by Eqs. (8.8) and (8.9) 
can be rewritten as 



Re Feu, + ReF L < 0 
Imy out + ImK L = 0. 



(8.38) 
(8.39) 



364 OSCILLATORS 




(a) (h) 
FIGURE 8.20 Electrical circuits of parallel feedback oscillators. 



To obtain the analytical relationships between the active device and resonant circuit parameters, 
let us consider the simplified intrinsic MOSFET high-frequency small-signal equivalent circuit shown 
in Figure 8.19(b). The /-parameters of the equivalent circuit are 

Y u = jco (C gs + C gd ) F12 = -jcoC gi 4 
Y21 = gm- j® Cgd Y 22 = jco (C ds + C gd ) . 

An assumption of the lossless feedback elements allows us to provide a simple qualitative evalu- 
ation of the oscillator start-up conditions. For a Colpitts oscillator, whose basic circuit schematic is 
shown in Figure 8.20(a), the feedback and load admittances are Yi — jcoCi, Y 2 — l/(JcoL 2 ), and y L — 
l/R L + jcod. 

Then, by substituting all admittances in Eq. (8.35), the small-signal device transconductance g m 
required to excite the self-oscillations will be determined by 

1 C es + C, 

^ > T7^T^- (8 - 41) 

The self-oscillation frequency that depends on both transistor equivalent circuit parameters and 
feedback elements can be defined from 



1 Cg S + Cds + Ci + C3 

60 = Jt~~i ^ 7 \ • ( 8 - 42 ) 

^2 (C gs + C g d + Ci) (C ds + C 3 ) + (Cgs + Ci) Cgd 

If the value of the MOSFET intrinsic feedback capacitance C g d is negligible, the expression for 
the oscillation is simplified to 



1 

L2 \C es + Ci Cds + C3 



+ „ , „ ■ (8.43) 



From Eq. (8 .4 1 ) it follows that, at lower frequencies compared with the device transition frequency 
ft when the effect of the elements of the device equivalent circuit is not significant, in order to provide 
the soft build-up of the oscillation, it is necessary to choose the feedback elements, load resistance, 
and dc bias point to provide 

1 Ci 

Sm>— A (8.44) 

«L C 3 



SERIES FEEDBACK OSCILLATOR 365 



Because it was assumed that the value of the feedback susceptance S 3 = lmY 3 is positive, 
consequently, to satisfy the start-up and steady-state oscillation conditions, it is necessary to use the 
capacitance C3. By rewriting Eq. (8.43) as 



Cos + C\ 



2 L 2 (C gs + d) - 1 



(8.45) 



it is easy to show that when 



(C gs + Ci) 



(8.46) 



then the value of 6 3 becomes negative. This means that such an oscillator feedback element must be 
inductive with the value of L 3 = — l/(a>B 3 ). The circuit configuration for this oscillator is shown in 
Figure S. 20(b). 



8.5 SERIES FEEDBACK OSCILLATOR 

A generalized equivalent circuit of the series feedback oscillator is shown in Figure 8.2 1 (a), where Zj , 
Z 2 , and Z3 are the feedback elements. To calculate the steady-state operation mode, it is convenient 
to add the matrix of the series feedback element Z 2 and the matrix of the active device Z-parameters 
according to 



[Z] + [Z 2 ] = 



Z u + Z 2 
Z 2 i + Z 2 



Z12 + Z 2 

z 22 + z 2 



(8.47) 




[A 



(a) 




(b) (c) 
FIGURE 8.21 Equivalent circuits of series feedback oscillator. 



366 OSCILLATORS 



In this case, a system of two equations for the two-port network input current / in , output current 
/ out , input voltage V in , and output voltage V ollt can be written as 



Kb = (Z12 + Z 2 ) l m + (Z n + Z 2 ) Z 0 , 
V 0M = (Z 21 + Z 2 ) / m + (Z 22 + Z 2 ) / 01 



(8.48) 
(8.49) 



Since, for the oscillator shown in Figure 8.21(a), the boundary conditions are obtained in the form 



Vin — — Zl/m 
V 0 ut — — Z^/out 



the following matrix equation can be written as 



Zn + Zi + Z 2 Z[ 2 + Z 2 
Z 2 i + Z 2 Z 22 + Z 2 + Z3 



Thus, the steady-state oscillation condition can be expressed by 



= 0. 



Zn + Zi + Z 2 Zi 2 + Z 2 
Z 2 i + Z 2 Z 22 + Z 2 + Z3 



0. 



After some simplifications, Eq. (8.53) can be rewritten as 

(Zi 2 + Z 2 )(Z 21 + Z 2 ) 



Z 22 + Z 2 + Z 3 



Z n + Zi+Z 2 



(8.50) 
(8.51) 



(8.52) 



(8.53) 



(8.54) 



As a result, the steady-state oscillation condition for the series feedback oscillator represented as 
a one-port negative resistance oscillator can be written as 



Z 0 ut + Zl = 0 



where Z 3 = Z L , and 



Z 22 + Z 2 



(Z 12 + Z 2 )(Z 21 + Z 2 ) 
Zn + Zi + Zz 



(8.55) 



(8.56) 



Consequently, the separate equations for real and imaginary parts of the negative resistance 
oscillator, which are similar to Eqs. (8.36) and (8.37) for the parallel feedback oscillator corresponding 
to the steady-state oscillation process, can be obtained by 

ReZ 0Ut + ReZ L = 0 (8.57) 
ImZ out + ImZ L = 0. (8.58) 

Similarly, the start-up conditions for the series feedback oscillator can be rewritten as 

ReZ 0Ut + ReZ L < 0 (8.59) 
ImZ 0Ut + ImZ L = 0. (8.60) 



Let us obtain the analytical relationships between the active device and resonant circuit parameters 
of the oscillator based on the MOSFET device, the admittance F-parameters of the equivalent circuit 



SERIES FEEDBACK OSCILLATOR 367 



of which are given by Eq. (8.40). The ratios between admittance K-parameters and impedance 
Z-parameters can be expressed in the form 

Y 2 2 *12 ^21 ^11 
Zn = Zi 2 = Z 2 1 = Z9? — ■ (8.61) 

AY AY AY AY 

Then, the steady-state condition for the series feedback oscillator can be written as 
1 + Z 2 (Y n + Y l2 + Y 2l + Y 22 ) + Z x (Y n + Z 2 AF) 



K 22 + (Z 1 + Z 2 )Ay 



+ Z 3 = 0 (8.62) 



where Ay = 1/AZ and AY — Y n Y 22 — Y 12 Y 2l . For a simple oscillator circuit configuration with 
external gate inductance shown in Figure 8.21(£>), the feedback elements including load are defined 
as Z] = j(oLi, Z 2 = \ljo)C 2 , and Z 3 = /f L . 

Usually, in a wide frequency range up to 0.3/t, it is possible to neglect the intrinsic gate-drain 
capacitance C g d without the substantial loss of accuracy. Then, the small-signal transconductance g m 
corresponding to the soft start-up condition and the frequency of the self-oscillations can be evaluated, 
respectively, by 



Cds + c 2 



1 



c ... 



Cds + c 2 



(8.63) 
(8.64) 



From Eq. (8.63) it follows that the oscillation build-up can be easily provided by choosing the 
MOSFET device with the smaller drain-source capacitance Cds and using a sufficiently large value 
of the feedback capacitance C 2 . 

In a common case, different sets of two-port network parameters can be used for oscillator design 
including impedance Z-parameters, admittance /-parameters, or scattering 5-parameters. The choice 
of any type of these parameters depends on the operating frequency, required design accuracy and 
implementation technique, availability of the small- or large-signal active device parameters, or the 
possibility of using proper measurement and simulation tools. For example, in terms of one-port 
negative resistance approach, the conditions for self-oscillations expressed through the small-signal 
5-parameters can be written as 

K = 1 + |A|2 - |5 " |2 - |fe|2 < 1 (8.65) 
2IS12S21I 

r m r s = 1 (8.66) 
r 0U ,r L = 1 (8.67) 



where K is the stability factor, A = SnS 22 — S\ 2 S 2 i, T in and T s are the input and source reflection 
coefficients, r ou( and Tl are the output and load reflection coefficients, respectively [14]. The stability 
factor should be less than unity for any possibility of self-oscillations. The passive terminations Ts 
and T L must be added to resonate input and output ports at the oscillation frequency. It should be 
noted that the conditions described by Eqs. (8.66) and (8.67) are inter-related, and if one condition 
is satisfied, then the other condition should be satisfied as well. The large-signal S-parameters of the 
transistor optimized for maximum output power can be measured by varying the input drive level and 
load impedance, or injecting signal into the output port [15]. By using the measured 5-parameters, 
the required ratios of terminal voltages and currents can be calculated. 



368 OSCILLATORS 



By converting 5-parameters to Z-parameters, the active device impedance parameters can be 
obtained as 



Z21 Z22 



(l-S n )(l-S 22 )-S l2 S 2 



(l + S u )(l-S 22 )+S 12 S 21 2S l2 

25 2 i (l-5 11 )(l + 5 22 )+Si25 21 



(8.68) 



that allows us to calculate the input or output impedance of the loaded active device in the paral- 
lel feedback two-port network, or negative resistance series feedback one-port network oscillator 
configurations. 



8.6 PUSH-PUSH OSCILLATORS 



Using a common collector configuration of the transistors and a series resonant circuit located between 
the devices in the push-pull oscillators simplifies the load connection and allows the operation 
condition at twice the operating frequency. Figure 8.22(a) shows the general simplified equivalent 
circuit of the balanced oscillator having two load resistors, one connected in parallel to the series 
resonant circuit inductor and the other connected into the inductor center point. The series inductor 
located between the two active devices is common for both oscillators. 

For a simplified circuit analysis, let us represent the oscillator schematic in the form of a general 
negative conductance oscillator with two active devices connected to a common two-port network, 
as shown in Figure 8.22(£>). The two-port network is characterized by the admittance K-parameters 
and terminal voltages represented by voltage phasors given as 

Vj = ViexpO,) V 2 = V 2 exp(j<h) (8.69) 

where V\, V 2 , (pi, and <p 2 are the magnitudes and phases of two voltage phasors, respectively. 




V 



J 



FIGURE 8.22 Simplified circuit schematics of bipolar push-pull oscillator. 



PUSH-PUSH OSCILLATORS 369 



^inl 


v," 




"I'll 






"vr 


Ym2 


V 2_ 




?21 


122. 




. V 2_ 



The relationships between the circuit currents and voltages in a steady-state operation mode 
through the admittance parameters in a matrix form can be written by 



(8.70) 



where K in i and Y in2 represent the input admittances of the negative conductance devices. 

Since the oscillator consists of two identical nonlinear active halves when V\ — V 2 — V and Vi n i — 
V- m2 — Vi„ and symmetrical passive linear two-port network when Y n — Y 22 and Y l2 — Y 2i , matrix 
Eq. (8.70) can be rewritten in the form of two equations as 



-Y in = Y n + y 12 expO» 
-Y m = Y 12 exp(-j<P)+Y n 



(8.71) 
(8.72) 



where 4> — <p 2 — 4>i is the phase difference between voltage phasors. 

Simplified analysis of the steady-state operation modes shows that, in such a balanced oscillator, 
there may exist two modes with equal amplitudes [2,16]: 



odd mode with 



even mode with 



= (2k+l)n where /t = 0, 1, 2, 



4> — 2 kit where k = 0, 1, 2, 



The frequency and amplitude of each of the two oscillation modes can be determined by solving 
the following equation: 



Yin(V,m) + Y n (aj)±Y 12 (co) = 0. 



(8.73) 



In a time domain, assuming symmetrical current waveforms flowing into the base of both transis- 
tors, their Fourier series expansions in a common case can be written by 



'i 



/f'cos cot + cos 2mt + /i 1 ' cos 3cot + • 



+ cos ncot 



(8.74) 



i 2 = if cos {mt + <p) + I 2 2> cos 2 (cot + <p) + /f cos 3 (cot + <p) + h if cos n (cot + cj>) 

(8.75) 

where n is the harmonic number. 

Consequently, in the odd mode, the currents are flowing in opposite directions providing a 
push-pull operation of both transistors having 180° out-of-phase base-emitter voltages. In this case, 
the circuit becomes antimetric and, at its center point, a virtual ground will be formed at the funda- 
mental frequency /o with zero fundamental voltage at this point. The output current flowing into the 
load 2/ff° is a result of the out-of-phase summation of currents given by Eqs. (8.74) and (8.75) as 



( <0) = 2h cos cot + 2h cos 3cot H h 2I 2k+l cos (2k + 1) cot 



(8.76) 



which contains only odd current components, and there are no odd components flowing into the 
load Rl (c) - At the same time, the output current flowing into the load ^L (e) is a result of the in-phase 
summation of currents as 



( <e) = 2/ 2 cos 2cot + 2/4 cos Acot + ■ ■ ■ + 2/ 2 k cos 2kcot 



(8.77) 



which contains only even current components. 



370 OSCILLATORS 




FIGURE 8.23 Equivalent oscillator circuits with (a) odd and (b) even modes. 



In the even mode, the currents are flowing in the same direction providing a push-push operation 
of both transistors having in-phase base-emitter voltages. In this case, the circuit becomes symmetric 
with a load resistor at the center of symmetry. For such an operation, the odd current components 
obtained by Eq. (8.76) will flow to this load, whereas the even components obtained by Eq. (8.77) 
will be dissipated at the load 2i?£_ . 

The equivalent circuits of the oscillator under odd and even operation modes are shown in 
Figure 8.23. For oscillations to occur in the odd mode, the negative conductance generated by the 
oscillator, whose schematic is shown in Figure 8.23(a), should exceed the resistive losses in the 
resonant circuit, that is 

Rey in + ReK^ 0) < 0. (8.78) 

For oscillations to occur in the even mode, the negative resistance generated by the oscillator, 
whose schematic is shown in Figure 8.23(£>), should exceed the resistive losses in such a resonant 
circuit according to 

ReZ in + ReZ[ e) < 0. (8.79) 

Generally, both odd and even operation modes can exist in the oscillator simultaneously depending 
on the values of the load resistances R^ and R^ \ Under a start-up condition given by Eq. (8.78), the 
even mode will be stable for the case of /?£° — 00 corresponding to the situation of the summation of 
the output powers of individual oscillators with resistive coupling [17]. A simplified criterion of the 
odd mode stable operation for the oscillator circuit shown in Figure 8.22(a) can be obtained as 



2*l 



M 



1 + 



2< e) 



.80) 



PUSH-PUSH OSCILLATORS 371 




(c) 



FIGURE 8.24 Simplified circuit schematics of bipolar push-push oscillator. 



when the odd components will flow into the load 2i?£° , while the even components will dissipate at 
the load R^. To improve stability of the odd operation mode, it is necessary to inhibit the oscillation 
condition given by Eq. (8.79). 

Such a balanced oscillator configuration with a series resonant circuit creates a simple opportunity 
to double the oscillation frequency when each half-circuit oscillates at the resonant frequency /o, 
while the output signal at the load oscillates at the frequency 2/ 0 . In this case, it is necessary to 
provide stable operation in the odd mode and to inhibit the oscillations in the even mode by removing 
the load resistor \ This results in the oscillator circuit configuration shown in Figure 8.24, where 
the variable capacitors C v can be used for frequency tuning. For a lossless tank inductor L, the start-up 
amplitude oscillation conditions can be rewritten as 

ReZ,„ < 0 (8.81) 
ReZ in + 2R L > 0 (8.82) 

when the even harmonic components dissipate at the load R^. A significant margin in negative 
resistance can be achieved compared with an equivalent single-ended oscillator, even at very high 
frequencies [18]. The outputs for the out-of-phase odd components can be connected to the device 
emitters. Because, for the even output current components, the transistors are operated in phase, it is 
called the push-push operation mode. A push-push operation is the inverse to a push-pull operation, 
in that the load is either conductively or capacitively coupled to the center point of inductor or 
transmission line located between the bases (gates) of the active devices [19,20]. In this case, the 
currents flow into the load in the same direction during both half periods or 180° phases of the active 



372 OSCILLATORS 



null 
point 




2/o 

FIGURE 8.25 Push-push and balanced microstrip oscillator configurations. 



device operation. Since there are two 180° phases during each cycle, the circuit acts as a frequency 
doublet'. Such a push-push oscillator is a result of its odd operation mode with virtual ground; 
therefore, ideally this design is independent of the output load pulling compared to a fundamental 
oscillator approach. 

At microwaves, the resonant circuits usually incorporate the transmission lines as inductive el- 
ements or resonators. Figure 8.25 shows the push-push oscillator using half wavelength or A/2 
microstrip resonator [21]. For the fundamental frequency f 0 , this resonator has a null point at the 
center of microstrip line, being a point of the oscillator symmetry, which is considered as a virtual 
ground or short-circuited point. In this case, the resonance voltage has maximum values at both 
ends of the resonator with a phase difference of 180°, and the resonance voltage is zero at the center 
of the resonator. For the desired second-harmonic frequency 2/o, such a point could be considered 
as an open-circuited point. The null point is an ideal output port to combine the second-harmonic 
signals from both active devices effectively. To compromise the maximum second-harmonic output 
signal and significant harmonic suppression, a sufficiently small series capacitor is usually used. The 
operating bias points with zero gate voltages can provide nonlinear operation of the used high electron 
mobility transistor (HEMT) devices generating a high-power second-harmonic signal. The impedance 
of half-wavelength microstrip open-circuited resonator is optimized to make the impedance of the 
output port close to 50 Q. For such a microwave push-push oscillator using two Fujitsu FHX35LG 
devices, a maximum power of 8.4 dBm at the second-harmonic frequency of 21.68 GHz with the 
fundamental frequency suppression of 26 dB was achieved. 



8.7 STABILITY OF SELF-OSCILLATIONS 

Applying dc bias to the active device does not generally result in the negative resistance or proper 
feedback condition. This condition has to be induced in these devices, and it is determined by the 
physical mechanism in the device and chosen circuit topology. The transistor in the oscillator circuits 
is mostly represented as the active two-port network, whose operation principle is reflected through 
its equivalent circuit. The influence of the circuit and transistor parameters can result in the hysteresis 
effect or oscillation instability in practical design. In high-frequency practical implementation, the 
presence of the parasitic device and circuit elements can contribute to the multiresonant circuits. 

A steady-state free-running oscillation build-up is provided with the velocity due to dissipation 
factor 5 = [R oul (I) + RlVL for a single series resonant circuit with an inductance L and 8 — [G ou t(V) + 



STABILITY OF SELF-OSCILLATIONS 



373 



-V|.(/,w) 



^1 



«l(A ») 



G ollt (I% «) 



T 
X 



T 
X 



FIGURE 8.26 One-port (a) negative resistance and (b) negative conductance oscillator circuits. 



Gl]/C for a single parallel resonant circuit with a shunt capacitance C shown in Figures 8.26(a) and 
8.26(b), respectively, the value of which is reduced with the decrease of negative output resistance 
RaatU) or conductance G 0Ut (V). It becomes zero in a steady-state oscillation mode when R ont (r) + 
R L — 0 or Gom(V) + Gl = 0, which is written in a general immittance form as 



ReW ou ,(A 0 ) + Reiy L = 0 



(8.83) 



where A 0 represents a steady-state amplitude of the self-oscillation, voltage or current. 

A stability criterion for such simple oscillator circuits, with only an amplitude dependence of 
output resistance or conductance, can be easily calculated through the perturbation method. Let us 
consider small perturbation AA > 0 in the operating point ReW(Ao) when A — Aq + AA. Using the 
linear Taylor series expansion results in 



ReW(A 0 + AA) = ReW(A 0 ) + 



aReW(Ao) 
3A 



AA. 



(8.84) 



A steady-state oscillation mode will be stable if an amplitude of the oscillation dissipates according 
to exp(— St) that is obtained from the solution of a second-order differential equation, describing the 
electrical behavior of each single resonant circuit shown in Figure 8.26 [2]. This takes place when 
ReW(A 0 + AA) > ReW(A 0 ). Therefore, a stability criterion is 



3ReW(A 0 ) 
3A 



> 0 



(8.85) 



which means that, as the active device negative resistance or conductance reduces with increase of 
the oscillation amplitude, stable oscillations are established in the oscillator. 

In a common case of (a) negative resistance or (b) negative conductance one-port transistor oscil- 
lator circuit model shown in Figure 8.26, a complex equation consisting of the nonlinear immittances 
in the steady-state stationary operation mode can be expressed as 



W(A 0 , w 0 ) = Re W (A 0 , co 0 ) + jlmW (A 0 , w 0 ) = 0 



(8.86) 



374 OSCILLATORS 



where 

ReW(A 0 , coo) = ReW 0Ut (A 0 , m 0 ) + RsW L (A 0 , w 0 ) (8.87) 
ImW(A 0 , m 0 ) = ImWou, (A 0 , o) 0 ) + ImW L (A 0 , ft*,) . (8.88) 

Assuming a solution of Eq. (8.86) as a(t) — A 0 cosc<;o t or a(t) — AoRe[exp(/&>o tj], one can find 
the amplitude Ao and frequency coq of free-running oscillations. Then, consider the small perturbation 
from the steady-state stationary mode when the amplitude and frequency of the oscillation are turned 
out to be A = A 0 + AA and m — m 0 + Am, respectively, where AA A 0 and Aw <3C o>o- Under 
conditions of small perturbations from the steady-state mode, a nonlinear system behaves as linear, 
and amplitude of the oscillation dissipates exponentially. Therefore, the expected solution can be 
written as 

a{t) = (A 0 + AA)exp(-A5f)cos(c!Jo + Aco)t (8.89) 



a(t) = (A 0 + A A) Re [exp j (co 0 + Aco + jAS)t] . (8.90) 

According to Eq. (8.90), the steady-state oscillation condition in the form of Eq. (8.86) can be 
rewritten as 

ReW(A 0 + AA, co 0 + Aco + jAS) + jImW(A 0 + AA, co 0 + Aco + j AS) = 0. (8.91) 

By applying the linear Taylor series expansion of Eq. (8.91) by degrees of the small parameters 
AA, Aco, and AS for each components and replacing one complex equation with two equations for 
real and imaginary parts, the following system of two equations in consideration of Eq. (8.86) can be 
written: 

3ReW dReW 3ReW 

AA+ Aco+ AS = 0 (8.92) 

3A dm dS 

dlmW aimW dlmW 

AA+ Am+ A,5 = 0. (8.93) 

9A dm dS 

It is known that for analytic function W(m + jAS) = ReW(m, AS) + jlmW(m, AS), the 
Cauchy-Riemann equations can be written as 

3ReW BlrnlV 

= (8.94) 

dm dS 

dReW dlmW 

= . (8.95) 

35 dm 

Then, excluding Am from Eqs. (8.92) and (8.93) and using Eqs. (8.94) and (8.95) result in 

3ReW 3ImW 3ImW 3ReW 



^ dA dm dA dm , Q 0 a\ 

AA 



\ dm ) + V dm ) 



STABILITY OF SELF-OSCILLATIONS 



375 




ib) 

FIGURE 8.27 Parallel feedback (a) electrical and (b) equivalent oscillator circuits. 



The steady-state stationary mode will be stable if a small perturbation of the oscillation amplitude 
AA dissipates in time and the values AA and A<5 have the same signs. Thus, the stable oscillations 
are established in the oscillator if 

dReW dlmW dlmW dRsW 

> 0 (8.97) 

9A dco dA dm 

which corresponds in terms of immittance parameters to a general form of a stability condition 
obtained by Kurokawa [22]. 

Figure 8.27 shows (a) the electrical and (b) the equivalent circuits of the parallel feedback negative 
resistance oscillator with a common collector. In this case, a transistor has been configured and biased 
so that the output resistance is negative. The capacitors Ci and C2 are used to increase a regeneration 
factor and to provide steady-state oscillation conditions of the amplitude and phase. 

In this series circuit configuration with the negative imaginary part of output transistor impedance, 
the load is constant and the negative output resistance R out — ReZ 0U1 (/, m) decreases with increase of 
the oscillation frequency. As a result, 

dX d / 1 \ 1 

— = — [toL - — - )=L + -— > 0 (8.98) 

and 

3R d dR M 

— = — (i? 0 ut + Rl) = > 0. (8.99) 

00) dco dco 

Consequently, by taking into account Eq. (8.97), to obtain stable oscillations in a steady-state 
operation mode, it is required to satisfy the following sufficient conditions for the series resonant 
circuit negative resistance oscillator with constant load: 



(8.100) 



376 OSCILLATORS 



8.8 OPTIMUM DESIGN TECHNIQUES 
8.8.1 Empirical Approach 

For optimum oscillator design with maximum output power, generally it is required extensive small- 
and large-signal measurements. The small-signal 5-parameter measurements can be made at several 
frequencies, along with estimated device equivalent circuit parameters, including package para- 
sitics. Then, a computer optimization program is used to match the measured 5-parameters to the 
5-parameters computed for the active device from its equivalent circuit. The next step is to vary 
those elements of the equivalent circuit, which can vary under large signals. By varying the active 
device nonlinear equivalent circuit parameters, the set of large-signal 5-parameters can be obtained 
corresponding to the saturation condition where maximum oscillator output power can be achieved. 

The simplified design approach assumes that all of the 5-parameters except the magnitude of 
52i are constant under large signals. For example, for a metal-semiconductor field-effect transistor 
(MESFET) device up to saturation with small limitation in accuracy, it is possible to derive the large- 
signal behavior of the main device nonlinear elements as functions of the device transconductance g m 
[23]. Consequently, the 5-parameters become functions of g m only and, at each incremental reduction 
of g m , are recomputed and optimized along with power gain. If one were interested in the 1-dB 
compression point, the 5-parameters used would be those at that point. 

For a common source (emitter) power oscillator, it is helpful to characterize the maximum power 
through the saturated output power P sat and small-signal transducer power gain Gt of the correspond- 
ing power amplifier. An empirical expression for output power of a common source MESFET power 
amplifier can be written by 



exp 



G T P n 



(8.101) 



where P m is the input power [23]. 

The objective is to maximize the oscillator output power P osc = P out — P- m by providing a condition 



d/'osc d(/ , out — P m ) 

~dp~ ~ d~^ 

Substituting Eq. (8.101) into Eq. (8.102) yields 



= 0. (8.102) 



dPi 

From Eq. (8.103) it follows that 



d^oul _ / GjPin 

= G T exp 1 = 1. (8.103) 



ex P ( ) = G t 

r cat 



Pm = InGx 
Psat Gt 



(8.105) 



As a result, 



P«* I 1 - ) (8.106) 



G 



OPTIMUM DESIGN TECHNIQUES 377 



and the maximum output power of the oscillator can be approximated by 




(8.107) 



Consequently, the oscillator maximum power can be predicted from the saturated power and 
small-signal transducer power gain of a common source power amplifier. From Eq. (8.107) it follows 
that the oscillator output power P osc will approach P sat at low frequencies where the small-signal 
transducer power gain Gt is large. As the transducer power gain approaches unity, the oscillator 
output power approaches zero. 

In this case, the maximum efficiency power gain Gme can be calculated from 

G T - 1 

G ME = -^— ■ (8.108) 
In G T 

For example, a MESFET device, which is characterized by the small-signal transducer power gain 
Gt = 7.5 dB and saturated amplifier output power of 1 W, would be capable of a maximum oscillator 
power of 515 mW. The maximum efficiency power gain at this point calculated from Eq. (8.108) is 
equal to Gme — 4.3 dB. The expression for maximum efficiency power gain Gme given by Eq. (8.108) 
can be used to determine at what gain level the large-signal 5-parameters were to be used for optimum 
oscillator design. 

A quasilinear design approach based on the measured small-signal 5-parameters and static output 
voltage-ampere device characteristic can be used to predict the oscillator output power [24] . However, 
this method assumes that the significant nonlinear effects in GaAs MESFET represent the rapid 
increase in the gate-source conductance due to the forward-biasing effect of the gate-source Schottky 
diode and output drain-source conductance [25]. To maximize the output power of a series feedback 
MESFET oscillator, an analytic procedure was derived using the input and output fundamental 
voltages as independent variables. The systematic approach to oscillator design using large-signal 
5-parameters, which may be measured under high drive conditions or obtained through the use of an 
active device nonlinear model, represents an alternative method [26]. The device can be modeled in a 
quasilinear approximation by its large-signal 5-parameters, each assumed to be a function of a single 
variable when 5n and 52i are the functions of the input fundamental voltage amplitude, whereas 5i2 
and 522 are the functions of the output fundamental voltage amplitude. As a result, the set of four 
equations describes the oscillation condition that requires standard computer-based nonlinear root- 
finding methods to determine the optimum feedback parameters providing a delivery of maximum 
output power into the load. 

A technique for the design of microwave transistor oscillator, in which measurements made on an 
experimentally optimized power amplifier, has been presented in [15]. Generally, the transistor power 
amplifier is easily analyzed and optimized than the corresponding oscillator because, in quasilinear 
approximation at a given frequency and bias point, only two parameters need to be varied: the input 
RF drive level and the output load impedance. Once these parameters are experimentally optimized, 
measuring the input impedance and constructing an input matching circuit complete the design. In the 
case of the transistor oscillator, however, there are a multitude of possible oscillator configurations 
and a large number of interacting circuit elements, which must be varied to optimize an oscillator 
for maximum power. But since we know that a transistor operates under the same set of RF voltages 
and currents when delivering its maximum output power, it is possible to take information obtained 
from an easily optimized power amplifier and use this information to calculate optimum oscillator 
configurations. 

The first step in this procedure is to experimentally optimize the large-signal behavior of the 
transistor by varying the load impedance and drive level for the maximum output power delivered 



378 OSCILLATORS 



into the load. The incident and reflected waves a\, b\, a 2 , and b 2 shown in Figure 8.28 can be measured 
with calculation of the output power delivered into the load according to 



Pl = \bi\- 1 



\S'u 



S>yj 



I S" 



(8.109) 



where 



S'u = 



"1 



bi , b 2 
o 2 9 — ^21 — 

a-i ct\ 



S' n is the input reflection coefficient Ti and S' 22 is the output reflection coefficient T 2 [14]. 

Using these three measured parameters S' n ,S' 21 , and S' 21 , it is then possible to calculate the required 
ratios of the transistor terminal voltages and currents 



Yi 

h 
h 

Yi 
h 

Yi 
h 

h 



J 21 



+ 1 



J 2I 



l + S' n 

1 

^22 



I -S' n 
1-S'u 



S" 

''21 



+ 1 



1-5' 



1 



S 22 



1 + S'n 



(8.110) 

(8.111) 
(8.112) 

(8.113) 

(8.114) 



where Vi and Ii are the transistor input port voltage and current, V 2 and I 2 are the transistor output 
port voltage and current, and Rs is the source resistance considered as the characteristic impedance 
of the measuring system, as shown in Figure 8.28. 






l ai 































r, r 2 

FIGURE 8.28 Power amplifier schematic to be experimentally optimized. 



OPTIMUM DESIGN TECHNIQUES 379 



8.8.2 Analytic Approach 

Depending on a type of the transistor used, operating frequencies, required output power and spectral 
characteristics, most of the oscillator schematics can be reduced to the two basic arrangements with 
external positive parallel or series feedback shown in Figures 8.29(a) and 8.29(b), respectively. One 
model may be preferred over another, depending on the oscillator configuration and characteristics. 
The two-port network representing an active device can be generally characterized by a system of 
immittance W-parameters. This describes parameter systems of a two-port network, such as a system 
of admittance y-parameters for parallel feedback or a system of impedance Z-parameters for series 
feedback oscillator circuits, respectively. The oscillator circuit and the output load are characterized 
by immittances Wi, W 2 , and W3 — Wl, respectively. 

Then, the steady-state oscillation condition for a single frequency of oscillation can be expressed as 

W oul +W L = 0 (8.115) 

where the active two-port network, together with the feedback elements W\ and W 2 , will be 
considered as a one-port negative resistance oscillator circuit. 

To optimize the oscillator circuit in terms of the maximum value of the negative real part of the 
equivalent one-port network output immittance, the expression for output immittance W out should be 
written as 

W °* = W " +W2 ~ W ll + W 1 + W 2 (8 - 1 16) 

where the minus signs in the factors correspond to output admittance of the circuit shown in Fig- 
ure 8.29(a) and the plus signs correspond to the output impedance of the circuit shown in 
Figure 8.29(b). Such an optimization approach was first applied to a bipolar transformer-coupled 
oscillator to define maximum output power [27]. Later it was used to determine optimum feedback 
elements with reference to a series feedback MESFET oscillator using a simplified device equivalent 
circuit without the intrinsic feedback gate-drain capacitance [28]. 

It is convenient to represent this optimum approach in a general form regardless of the type of the 
active device, for commonly used parallel and series feedback oscillators [2,29]. The first step in the 
design is to determine the optimum combination of the values of the feedback reactive elements Im W\ 
and ImVK2, which maximize the absolute value of the real part of the output immittance ReW 0Ut at 
the desired frequency of oscillation. Such a condition will permit to obtain self-sustained oscillations 
with the largest amplitude that implies maximum output power delivered to the load [30]. Analyzing 



o- m -o 



-o- |Z| -o- 



FIGURE 8.29 Two-port oscillator circuits with (a) parallel and (b) series feedback. 



380 OSCILLATORS 



Eq. (8.116) in extremum, we can find the optimum values Imiyf and ImW 2 °, at which the negative 
value ReW 0lll is maximal, by solving 

3Reiy 0Ut 3ReW oul 

— =0 — =0. (8.117) 

dlmWi aimVK, 

As a result, the optimum values IrnVKf and ImW^ depend on the immittance parameters of the 
active device two-port network according to 



lmW? — T 

1 + Im(W 21 - W 12 ) 



ReOVi2 + W2i) " 
± Re ( Wu + Wi ) 



lm(W l2 +W 2l ) 
lmW n =f 

(8.118) 

y i i/o Re(W 21 + W l2 T2W 2 )Re(W 2l -W 12 ) ^lm(W 12 + W 21 ) 

imWX = ± ± . (8.119) 

2 2hn(W 21 -W u ) 2 

The next step is to determine the optimum load immittance W L in a steady-state operation mode 
defined by Eq. (8.1 15). By substituting expressions for ImW° and ImlV" intoEq. (8.1 16), the optimum 
real and imaginary parts of the output immittance W° ut will be respectively defined by 

Re 2 (W12 + W21 T 2Wi) + Im 2 (W 21 — W i2 ) 

ReW° = ReW 22 + ReW, ^ — — — (8.120) 

out 4Re(W n + Wi + W 2 ) 

ImW° = ImW 22 + ImW, 0 - Re ^ Wn ~ Wl2 \ e (W° - W 22 - W 2 ) . (8.121) 
out 2 Im(W 2 i - W 12 ) V ; 

In a large-signal operation mode, all immittance parameters generally become the functions of the 
voltage amplitudes across the elements of the device equivalent circuit. For the negative resistance 
(conductance) one-port oscillators shown in Figure 8.29, the output power can be written as P out = 
V 2 G 0Ut (V)/2 in terms of y-parameters or P 0M — / 2 i? out (/)/2 in terms of Z-parameters, where V is 
voltage amplitude across the load resistance and / is the amplitude of the output current flowing into 
the load. Since, for the same output voltage or current amplitude, the combination of the optimum 
feedback parameters obtained by Eqs. (8.118) and (8.119) provides a maximum negative real part 
of the output immittance according Eq. (8.120), then the maximum power will be delivered to the 
load. Thus, such an analytic approach presumes that, once the nonlinear model of the active device is 
developed, the elements of the optimum feedback parameters and the load can be easily and explicitly 
calculated under both the small-signal and large-signal conditions corresponding to the start-up and 
steady-state operation modes, respectively. 

An analytical evaluation of the start-up and steady-state oscillation conditions for a bipolar 
oscillator can be done using a simplified transistor equivalent circuit shown in Figure 8.30 [2]. In this 
case, the parasitic lead inductances and resistances can be considered amongst the external feedback 
circuit. The typical condition r n 3> l/(a>C n ) simplifies the analytical calculations substantially without 
a significant decrease in accuracy at high frequencies. Let us consider the transistor parasitic lead 
inductances L b , L e , and L c among the external feedback elements Zi, Z 2 , and Z L , respectively. In 
addition, loss in the feedback elements should be taken into account in elements and r e . 

As a result, the optimum values of imaginary parts of the feedback elements X° and X°, expressed 
through the parameters of the bipolar transistor equivalent circuit, can be calculated from 

1 a> 

X\=-—-r h — (8.122) 

1 0} 

x; = -— — -r.— . (8.123) 
ZcoC c cor 

where coj — g m /C n is the radian transition frequency. 



OPTIMUM DESIGN TECHNIQUES 381 




FIGURE 8.30 Simplified series feedback bipolar oscillator circuit. 



The real and imaginary parts of the optimum output impedance Z° ul = R° ut + jX° ut , including the 
collector series resistor r c and lead inductance L c , are given by 



Rnnt — r C + 



r b + r e + R n 



e + Rn aa>j: cjj a (^2a>C l 



(8.124) 



X° m = a>L e ~ T ^ + (R° M ~ r e ) — (8. 125) 

where Rn — a lgm and 

1 

a= 

/ (o 
1+ — 

From Eq. (8.124) it follows that, as frequency increases, the absolute value of the negative 
resistance R° M reduces and becomes zero at maximum oscillation frequency / max . Without accounting 
for the parasitic parameters of the bipolar transistor equivalent circuit, the expression for/ max is 

h 

J . (8.126) 
87rr b C c 

This coincides with the well-known expression for a maximum oscillation frequency / max of the 
bipolar transistor, on which a maximum available power gain becomes equal to unity and a steady- 
state oscillation condition can be established only for lossless feedback elements. Such a condition 
corresponds to the three-terminal mode of the transistor operation. However, the circuit may be 
capable of oscillating at frequencies greater than the/ max obtained by Eq. (8. 126), when it is operated 
in a two-terminal mode, by making Z\ an open circuit. In this configuration, no RF current is flowing 
into the base terminal to produce negative resistance (though the base terminal can still be used for 
biasing purposes). The transistor is now behaving as a two-terminal negative resistance, and the base 
resistance does not directly affect the maximum oscillation frequency, which will be determined by 
the parasitic resistances in the emitter and collector circuits [31]. 



382 OSCILLATORS 



TABLE 8.1 Microwave Bipolar Transistor Equivalent Circuit Parameters. 



Cc.pF 




C„, pF 


L e ,nH 


Z-b.nH 


L c , nH 


r e , Q 




r c , Q 


,/t, GHz 


0.5 


30 


40 


0.3 


0.3 


0.5 


0.3 


4.0 


1.75 


6.0 



The oscillator simulations were performed using a harmonic balance technique implemented in a 
Microwave Harmonica that is a part of the computer circuit simulator Ansoft Software, considering 
the oscillation frequency as an additional optimization variable [32]. The transistor equivalent circuit 
parameters are listed in Table 8.1. Accuracy of an analytic approach was verified based on a series 
feedback microwave bipolar oscillator shown in Figure 8.31. For a preliminary chosen oscillation 
frequency / = 4 GHz, the optimum oscillator feedback parameters according to the theoretical 
predictions given by Eqs. (8.122) and (8.123) must be equal to L = 1.2 nH and C — 0.8 pF. 

Figure 8.32 shows the oscillation frequency and output power in a steady-state mode for various 
values of the emitter bias resistor R e . In spite of the preliminary simplification, the simulation results 
obtained indicate minimum discrepancy between the chosen oscillation frequency and simulated 
oscillation frequency using an analytic calculation of the oscillator feedback parameters (taking 




OPTIMUM DESIGN TECHNIQUES 383 



into account the standard load resistance R L = 50 £2). An exact value of the oscillation frequency 
f — 4 GHz in a steady-state mode is realized for R e — 28 f2, when the output power P out is close to 
the maximum value. Moreover, the output power P ou , = 21 .5 dBm is very close to a maximum value 
of 21.9 dBm for optimum load R£ = 45 Q. 

A common gate MESFET oscillator circuit with a series feedback between the gate and the ground 
is shown in Figure 8.33. Such a circuit configuration was selected because of its inherent broadband 
negative resistance. If the correct feedback reactance is added, oscillations can occur from very low 
frequencies to approximately maximum oscillation frequency / max . Figure 8.33 also shows the small- 
signal MESFET equivalent circuit, which characterizes with good accuracy the device performance 
up to 50 GHz [33]. 

The results of microwave bipolar oscillator analytic design based on a quasilinear approach 
show the attractiveness and high effectiveness of preliminary analytical calculations of the oscillator 
feedback parameters according to the simple optimum analytical expressions. Therefore, one can 
assume that, to speedup the design process for microwave MESFET oscillators, a simplification 
of the analytical expressions for its feedback elements is possible [34]. For example, influence of 
the gate-drain capacitance C g d and transit time r can be ignored, and the transistor parasitic lead 
inductances L g , L s , and Ld can be considered among the external feedback elements Zi, Z%, and Zl, 
respectively. Then, the optimum values of imaginary parts of the lossless feedback elements X°, X\, 
and output reactance X° ut can be given by 



1 



L>Cds (^gs + ^g) + 



gm 

2<wCV 



X® = — Rds ( a>C<i S R s + 



x' 



2a>C„. 



(8.127) 
(8.128) 
(8.129) 



As a result, for an optimum series feedback MESFET oscillator, according to Eqs. (8.127) 
and (8.128), the optimum values of the reactances X° and X° should be inductive and capacitive, 




FIGURE 8.33 Series feedback microwave MESFET oscillator circuit. 



384 OSCILLATORS 

TABLE 8.2 Small-Signal Parameters of GaAs MESFET Equivalent Circuit. 



L g ,pH 


L s , pH Li, pH R g , Q fi gs , Q R s , Q R d , S2 


C gs , pF C gd , pF 


Cds, pF 


gm, mS 


50.4 


0.1 60.1 2.0 2.0 0.93 1.1 


1.2 0.087 


0.199 


97.4 



respectively. An analytical equation to calculate the optimum output resistance i?° ut in a small-signal 
operation mode can be written by 



Km = *. + 



R.i 



1 + (ft>C ds ^ds)" 



^d 



Rg ~\~ Re, 4" Ros \ 2(j!)Cg 



(8.130) 



To determine the differential drain-source resistance i?ds as a function of the optimum output 
resistance R° ut , it is enough to solve the quadratic equation for R is obtained from Eq. (8.130). As a 
result, 



where 



1 



l + yi-4(/C t -.R s )Gd 
2Gdso 



(8.131) 



Rg + Rs + R„ 



I 8m 
\ 2(dC c 



) 2 + « ul 



Rs) (wC ds ) 2 



Let us verify the accuracy of the analytic approach using the power microwave MESFET device 
with gate length / = 1 [im and gate width w — 4 x 200 |im. To determine a large-signal value of the 
output resistance R° M for a certain value of the load resistance R L , it is necessary to use the amplitude 
balance equation R° m + R$ + R^ = 0. In such a situation, i?ds is considered as a fundamentally 
averaged drain-source resistance R isl under large-signal operation. The nonlinear circuit simulation 
was performed for a 12 GHz microwave series feedback MESFET oscillator, with the small-signal 
parameters of its equivalent circuit listed in Table 8.2 [33]. 

According to the theoretical predictions, the optimum oscillator feedback parameters must be 
equal to L — 0.35 nH and C — 0.5 pF (for load resistance = 50 f2), as shown in Figure 8.34. In this 




FIGURE 8.34 Simulated series feedback 12 GHz MESFET oscillator. 



NOISE IN OSCILLATORS 385 




case, to satisfy the phase balance conditionX° ut + a>La + = 0, the value of the load reactance 
should be capacitive of Cl = 1 pF. A simulated value of the oscillation frequency is 10.72 GHz, which 
differs from the selected value by only 11%. Figure 8.35 shows the dependencies of the oscillation 
frequency / and output power P out on the load resistance R L From this it follows that maximum 
output power P out = 22.9 dBm can be obtained for load values in limits of 20-30 Q. 



8.9 NOISE IN OSCILLATORS 



The instantaneous output signal of an oscillator can be represented by 



o(0 ! 



1 + 



AA(t) 



cos [2nf 0 t + 0o + A0(f)] 



(8.132) 



where Aq is the voltage amplitude of the steady-state oscillations, fo is the oscillation frequency, <po is 
the initial phase of the oscillations at t — 0, AA(t) and A(p{t) are the amplitude and phase deviations 
of the corresponding amplitude and phase fluctuations, respectively. Generally, the nature of the 
fluctuations may be discrete or random where the discrete signals are called the spurious, appearing 
as distinct spectral components, while the random fluctuations are called the phase noise. 
The instantaneous frequency as a function of time can be written as 



f{t) = ^-4- {2jt f ot + <t>° + A <t> (*>] = fo + A/ (t) 
2tt at 



(8.133) 



where 



A/(f)< 



1 dA(p(t) 



lit At 

Since the process associated with frequency fluctuations Af(t) is stationary, then 



A(p (t) — lit \ Af (r)dr 



I 



(8.134) 



386 OSCILLATORS 



which is in general a nonstationary process. However, as the phase fluctuation process is sufficiently 
slow during the natural period of the oscillations, the stationary model to describe the phase noise 
performance of free-running oscillators can be used. It should be noted that an autonomous free- 
running oscillator does not have a reference plane and the initial phase <p 0 in an autonomous system 
can be chosen arbitrary, for example, of zero value. 



8.9.1 Parallel Feedback Oscillator 

It is very important for the oscillator noise model to express a clear relationship between the oscillator 
spectral noise power density, resonant circuit, and active device parameters. The simple Leeson linear 
model for a feedback oscillator, which was derived empirically, is based on the expectations that the 
real oscillator has two basic components [35]. The first component is caused by the phase fluctuations 
due to the additive white noise at frequency offsets close to the carrier, as well as due to the noise 
having a mixing nature resulting from the circuit nonlinearities. The second component is a result of 
the low-frequency fluctuations or flicker noise upconverted to the carrier region because of the active 
device nonlinear effects. 

The phase noise at the input of the power amplifier is added to a signal as the sum of every 
bandwidth A/ = 1 Hz, each producing an available noise power at the input of the noise-free 
amplifier, as shown in Figure 8.36(a). Maximum power delivery can be achieved when the source 
internal impedance is conjugate-matched to the input impedance of the amplifier. As a result, only 
one-half of the root-mean-square noise voltage e n appears across the amplifier input and is equal to 



e ta = y = = VfkTR (8.135) 

where F is the noise figure, k is the Boltzmann constant, T is the absolute temperature, and R 
is the equivalent resistance, which can be represented as the input resistance for the input root- 
mean-square noise voltage [36]. The input phase noise produces a root-mean-square phase deviation 
Ai^nns = A0/V2 at each offset frequency ±/ m from the carrier f 0 , as shown in Figure 8.36(£>), for 
which a total power-wise sum can be written for a small phase perturbation as 




A</> = A0 rms V2 = = , /— (8.136) 



where V in = ^/2P itl R is the signal voltage amplitude at the power amplifier input. 




NOISE IN OSCILLATORS 387 



A<|> 



Phase 
module lor 



Noise-free 
amplifier 



5,,. (/„,') 

— o 

Output 



Rusonatoi 



(/>) 



FIGURE 8.37 Equivalent model of parallel feedback oscillator. 



As a result, the double-sideband spectral power density of the thermal phase noise in a frequency 
bandwidth A/ = 1 Hz can be written as 

FkT 

S^ = A0 2 =— . (8.137) 

"in 

The Leeson model consists of an amplifier with a noise figure F and a resonator (or filter) in the 
feedback loop, as shown in Figure 8.37(a) [36]. The oscillator phase noise is modeled by assuming a 
noise-free power amplifier and adding a phase modulator to its input. Based on empirical predictions, 
the phase noise level of the oscillator at an offset frequency f m from the carrier/o can be described by 

W.) = S«</.>yvY */.<=£- (8.138, 
S 0 (/m) = S Atjl (/m) for f m > (8. 139) 



where Sa</> (fm) is determined using Eq. (8.137) as 

FkT ( f c \ 

S A </,(/m)= — h + 7 1 ( 8 - 14 °) 

"in \ /m / 

taking into the effect of the signal purity degradation due to the low-frequency flicker noise effect 
close to the carrier, described empirically by the corner frequency / c . It should be noted that the 
empirical Leeson equation for S A ^ (f m ) contains a multiplication factor of two in the numerator. 
Moreover, accurate agreement was achieved between the model and experiment results when the 
power density S A ^ (f m ) was expressed in terms of the compressed (or large-signal) power gain G and 
output power P out as (f m ) = 2GFkT/P mt [37]. 

The parameter F in Eq. (8.140) is associated with the active device noise figure and can be 
called an effective noise factor because, generally, it should represent the effects of the active device 
noise sources and the cyclostationary noise resulting from periodically varying processes in practical 
oscillators. Due to the inherent nonlinear nature of the active device, the effects of intermodulation 
between the wideband white noise and various harmonics of fundamental frequency (for example, 
nonlinear transformation of the noise near the third harmonic downconverted to the near carrier region 
due to mixing effect with second harmonic) must be included [35]. Also, the effect of low-frequency 
noise modulation of the current, resulting in the reactance modulation of the input impedance of 
the circuit (for example, variation of the phase angle of the device forward transfer function versus 
emitter current), cannot be neglected [38]. Hence, it is impossible to calculate F accurately without 
taking into account the effect of the oscillator resonant circuit. Therefore, for such a linear model, 
the effective noise factor F as well as the corner frequency f c can be considered more like fitting 
parameters, based on measured data. 



388 OSCILLATORS 



The corresponding combined expression to calculate the normalized double-sideband phase noise 
power spectral density or the double-sideband noise-to-carrier ratio at the input of the feedback 
oscillator can be obtained from 



^0 ( fm) = 



FkT 



1 + 



1 + 



fo 



(8.141) 



which gives an asymptotic model showing generally the noise reduction of 9 dB/octave in the offset 
region with predominant low-frequency 1// noise, 6 dB/octave in the offset region due to feedback 
loop and 0 dB/octave representing the thermal or white noise spectrum. 

The single-sideband noise-to-carrier ratio at the input of the feedback oscillator can be described by 



1 FkT ( / c 

1 in \ Jm 



i + 



fo 



2Q L f n 



(8.142) 



whose idealized sideband spectral behavior for different values of the loaded quality factors is 
illustrated in Figure 8.38. For the Iow-Ql case, there are regions with 1//JJ and 1//^ dependencies 
for spectral power density close to carrier, as shown in Figure 8.38(a). For the moderate-gL case, 
Figure 8.38(2?) demonstrates only 1//^ dependence as far as intersection with thermal noise floor. 
For the high-QL case, the regions with 1 //^ and 1 // m dependencies are observed near the carrier, 
as shown in Figure 8.38(c). Closest to the carrier, 1//^ phase noise behavior is a result of random 
frequency modulation of the oscillator by low frequency \lf noise. In the region of 1 /f£ phase noise 
behavior, the white noise causes random frequency modulation. The 1 // m dependence is due to the 
mixing up of the \lf noise with the oscillation frequency. Finally, the phase noise becomes constant, 
which is a result of the mixing up of white noise around the oscillation frequency. 

To calculate the same phase noise power spectral density at the oscillator output, it is necessary 
to replace the input power P- m by the power available at the output P out and to multiply the numerator 
of Eq. (8.142) by the power gain G. As a result, neglecting the effect of flicker noise and considering 
the case of / m <C/o, one can obtain 

GFkT ( f 0 \ 2 

where 



1_ 2l 

Qo 



2 



(8.144) 



is considered the transducer power gain and Q 0 is the unloaded quality factor [39]. From Eqs. (8. 142) 
and (8.143) it follows that, to minimize the oscillator phase noise, it is necessary to reduce the noise 
figure F and to increase the input power P in (or the output power P out for a fixed power gain G of 
the amplifier) as much as possible. In addition, for frequency offsets inside the resonator bandwidth, 
it is desirable to maximize the oscillator loaded quality factor Q L . However, the resonator insertion 
loss and loaded £?l are inter-related, and one cannot arbitrarily increase Ql without increasing the 
insertion loss, otherwise a larger power gain G is needed. The two competing effects result in an 
optimum loaded Q L of approximately one-half the unloaded Q 0 and insertion loss of about 6 dB 
[40,41]. Thus, the minimum noise occurs when Ql/Qo — 0.5 resulting in 

2FkT ( fo V 

L(/m) = lu UebJ • (8 - 145) 



NOISE IN OSCILLATORS 389 




Note that the difference in the optimum noise performance predicted by different definitions of 
the output power (power dissipated in the resonant circuit or power available at the amplifier output) 
is small [39]. 

Figure 8.37(b) shows another representation of the Leeson model with a phase feedback loop. 
Suppose that the phase-noise modulation occurs in the oscillator active element as A(p(t), and it is 
necessary to define how the oscillator reacts to this internal noise. As it is known from transmission 
theory, the transfer function of a modulated high-frequency signal, passing through a bandpass filter, 
equals to the transfer function of the modulating signal passing through an equivalent low-pass filter 



390 OSCILLATORS 



prototype. Thus, the phase relationship resulting from the null-phase condition on the loop and from 
the filtering of <j>(t) by the selective filter can be written as 



f(t) + A0(O = <MO 



(8.146) 



where 



fit) 



/ 



>(t)h(r -t)dr = 4>(t) x h(t) 



(8.147) 



h(t) is the impulse response of the equivalent low-pass filter, and the asterisk denotes a convolution 
product. The integral in Eq. (8. 147) converges for nearly all samples of <p(t) provided that the filter is 
linear and time invariant, and that the stationary random process <p(t) possesses a finite second-order 
moment. 

In this case, the Leeson formula for the double-sideband phase noise power spectral density of the 
feedback oscillator given by Eq. (8.141) can be expressed in a more general form 



S4, (o> m ) : 



S&<p (&>m) 



[H(K)-I][fi'0%)-1] 



(8.148) 



where H(joj m ) is the equivalent low-pass transfer function and asterisk denotes the complex-conjugate 
value [42]. 

Thus, by representing the transfer function of the first order low-pass filter as 



H (j(i>m) : 



a + ja)„ 



(8.149) 



where a = a> 0 /2Q L is the half-bandwidth of the resonator and a) m = 2nf m , Eq. (8. 148) can be rewritten 

as 



((%) = Sa<£ (o>m) 



1 + 



(8.150) 



which is similar to Eq. (8.141). 

Now consider a case of the second-order low-pass filter based on the two coupled resonators 
having the transfer function 



H(ja> m ) : 



0-2 



1 



a 1 + J co m a 2 + jco m i _ + j 2(l > m 8 

where a — ^/a\OL2 and & — («i + U2) /2^/a\a2 [43]. In this case, Eq. (8.148) can be rewritten as 



(8.151) 



1 - 2 



(8.152) 



which shows the substantially better phase noise performance in the near vicinity of the carrier 
frequency for the case of loosely coupled resonators when S > 1 . However, since the available output 
power becomes low, it is necessary to provide post low-noise output signal amplification. 



NOISE IN OSCILLATORS 391 



Let us quantitatively compare both cases of the Leeson phase noise models using the first- 
and second-order low-pass filters in the oscillator feedback loop for the same arbitrary chosen 
technical data: 

• Oscillation (carrier) frequency fo — 2 GHz. 

• Offset frequency f m =10 kHz. 

• Oscillator resonant circuit loaded quality factor <2l = 10. 

• Noise figure of the active device F — 6 dB. 

• Input power delivered to the device P m — 10 mW. 

• Corner frequency for flicker noise f c — 3 kHz. 

Substituting these parameters into Eq. (8.142) for a single-sideband phase noise spectral density 
of the oscillator with the first-order low-pass filter results in 



For the case of the oscillator with two coupled resonators, let us assume that the loaded Q L of the 
second resonator is five times as much as the first one, that is ofi = 6.283 x 10 s and ce 2 — 1.2566 x 
10 s . Then, 



which clearly shows the significant phase noise improvement compared to the oscillator having the 
first-order low-pass filter in a feedback loop. 

Figure 8.39 shows the typical oscillator output power spectrum. The noise distribution on each 
side of the oscillator signal is subdivided into a large number of strips of width Af located at the 
distance / m away from signal. It should be noted that, generally, the spectrum of the output signal 
consists of the amplitude and phase noise components. Hence, to measure the phase noise close to the 
carrier frequency, one needs to make sure that any contributions of parasitic amplitude modulation to 
the oscillator output noise spectrum are negligible compared with those from frequency modulation. 
The single-sideband phase noise L(f m ) usually given logarithmically is defined as the ratio of a signal 



L(/J = 10 log( 1.05 128 x 10" 10 ) = -99.78 dBc/Hz. 



L(/ m ) = 10 log(2.918 x 10~ 12 ) = -1 15.35 dBc/Hz 



/\dBm 



P, 




/.(/,„). dBc/Hz 



A/= 1 Hz 



n 



fo ~fm fo fo + fm 



f 



FIGURE 8.39 Oscillator output power spectrum. 



392 OSCILLATORS 



power f SS Af in one-phase modulation sideband per bandwidth A/ = 1 Hz, at an offset f m away from 
the carrier, to the total signal power P s . 

Despite some limitations of the linear Leeson model when device is operated in a conduction 
angle large-signal mode and output signal is not purely sinusoidal, which have an effect on the active 
device noise factor and low-frequency flicker noise upconversion, such an approach gives a sense of 
the phase noise performance for oscillators with different resonant circuits. This applies especially, 
if the theoretical results can be supported by sufficiently accurate measurements of a loaded quality 
factor of the oscillator resonant circuit and simulations of the effective noise figure based on the 
modeled active device parameters and operation conditions. 

In addition, such a simple model indicates the basic factors and provides the design rules that are 
necessary to follow in order to minimize the oscillator phase noise: 

• Choose the resonator with maximum unloaded quality factor go and optimize the loaded quality 
factor g L of the oscillator resonant circuit by proper load coupling. 

• Maximize the output power P L delivered to the load by maximizing the RF voltage amplitude 
across the resonant circuit with limitations due to active device breakdown voltage and operation 
in the saturation mode. 

• Choose a device with the lowest noise figure F and corner frequency f c for low-frequency 
flicker noise. 

8.9.2 Negative Resistance Oscillator 

Now consider the equivalent circuit of a simple single-resonant negative resistance oscillator shown in 
Figure 8.40, where the available noise power is assumed to be totally from the active device. Here, R n 
is the equivalent noise resistance associated with active device noise sources, the negative resistance 
i?out, and the equivalent output capacitance C out represent the device negative output impedance, L is 
the tank inductance, and is the load resistance. The derivation of the power spectral density will 
be based on the fact that the available noise power in the active device is amplified in a frequency 
selective way, resulting at resonance in the output power P L being dissipated in the load resistance 
Rl [44]. This will happen in a steady-state condition when the values of the negative resistance and 
the load resistance are close to each other. 

Then, assuming that R L + AR — —R oul and defining the magnitude of the mean-square noise 
current flowing into load from the mean-square noise voltage source, we can write 




(8.153) 




Cm L 
J I o TY-Y-Y-X 



mil 



o 



FIGURE 8.40 Simplified negative resistance oscillator noise model. 



NOISE IN OSCILLATORS 393 



Equation (8. 153) can be rewritten in the normalized general form as 

1 5 



(8.154) 

AR\ 2 ,/(o w 0 \ R t 

- r ) +Ql[ 

R L J \co 0 co 



where coq is the radian resonant frequency and <2l is the oscillator loaded quality factor at the resonant 
frequency. 

By normalizing to the power P L dissipated in the load resistor R L , Eq. (8.154) can be rewritten 
through the spectral power densities by 

= — rr (8-155) 



AR\ - ( CO COn 

' + Q 2 ' 



2 

_ CU 0 \ 

R L J ~ L \co 0 co J 

where S A $ = 4kTR„/(R L P L ), S,/, — i^Rl/Pl is the power spectral density of the noise current across 
the load resistor R L and A/ = 1 Hz. 

Since at small offset frequencies co m — co — coq close to the resonant frequency, 

co 0)q ^ 2co m 

coo CO COQ 

then Eq. (8.155) can be rewritten as 

s* = ~, rr^T — ( 8 - 156 ) 

\Rl) \ lo 0 

which is similar to the power spectral density at frequency offsets close to the resonant frequency for 
the parallel feedback oscillator. Equation (8.156) represents a Lorentz function corresponding to an 
exponential decay of the autocorrelation function in the time domain [45]. 
Since the total output power delivered to the load is equal to P L , 



oo 



where 

AR coq 

Aco„ = 

Rl 2g L 

is the Lorentzian linewidth (half-width at half-maximum), which is an oscillator spectrum linewidth 
characterized by the natural phase fluctuations due to the thermal and shot noises of the oscillator. 
However, in a common case, due to the variation of the oscillator resonant circuit parameters, flicker 
noise, pushing or pulling effects, the effective spectrum linewidth widens, especially close to the 
resonant frequency. 

By using a widely used definition of the loaded quality factor of the passive resonator in the form 



(8.158) 



394 OSCILLATORS 



where Am 3lSB is the full linewidth at half-maximum level, one can write 

2Aa)„ = — Aa) 3iB = kT — —^—. (8.159) 

As a result, there is a complete analogy between Lorentzian linewidth defined by Eq. (8. 159) and 
the expression for semiconductor laser homogeneous linewidth [46]. In this case, the characteristic 
energy kT corresponds to the photon energy, the oscillator output power P L corresponds to the 
laser output power, and the oscillator noise-gain ratio R„IR^ corresponds to the inversion factor 
representing the ratio between spontaneous emission rate and the optical gain rate. The widening of 
the oscillator spectral line due to the low-frequency fluctuations is similar to an inhomogeneous laser 
line broadening due to the Doppler effect. 

The normalized power spectral density can be expressed through the Lorentzian linewidth as 

2Aa>„ 2Aw.. 

S * = a 2,2 = (8 .160) 

Ami + < < 

showing a simple linear relationship between Lorentzian linewidth and oscillator phase noise spectrum 
at offset frequencies co m 3> Aa>„. Substituting Eq. (8.159) into Eq. (8.160) and taking into account 
that F — RJRl result in the single-sideband noise-to-carrier ratio 

kTF ( f 0 \ 2 

L(/ m )= — — (8.161) 

J 2P L \Q L f m J 

which is similar to the Edson noise formula [44,47]. 



8.9.3 Colpitts Oscillator 

As an example, let us define the linear phase noise model for a popular Colpitts oscillator, the 
simplified circuit schematic of which is shown in Figure 8.41(a). The power loss in the tank inductor 
L is included in the load resistance R L . The transistor equivalent circuit with voltage and current noise 
sources in shown in Figure 8.41(£>), where the resonator noise is modeled by a noise current ; n R. The 
noise voltage and current sources can be given through their mean-square values as 

-T- AkTAf 

4 = (8-162) 
v[ b = 4kTr b Af (8.163) 

(8.164) 

l[ c = 2qI c Af. (8.165) 

where the input noise current source is related to shot noise at the emitter-base junction due to 
electrons that recombine with holes inside the neutral base and (J. represents the shot noise generated 
at the collector-base junction due to collector electrons. The mean-square values of these noise 
sources in a bandwidth Af are given by 

7[ b = 2qI b Af = 2kTn bg7Z Af (8.166) 



;2 



2qI c Af = 2kTn cgm Af. (8.167) 



NOISE IN OSCILLATORS 395 



R.. 

wv 
I 




=FC 



"0- 
WV 



-o— 0 — 



\ 

v 



'" b (D r " > c " t K 0 ^ 



FIGURE 8.41 Equivalent circuits of Colpitts oscillator. 



where g % = Vr n , I b and / c are the dc base and collector currents, « b and n c are the ideality factors of 
the emitter-base and collector-base junctions, respectively [48]. 

The admittance K-parameters of the internal transistor excluding the base resistance r b can be 
obtained as 



i 



+ jco(Cj, + C c ) —jcoC c 



ja>C c 



jcoC c 



(8.168) 



where g m — filr n is the device transconductance. 

For a Colpitts oscillator with parallel feedback capacitors Ci and C2, it is convenient to represent 
all noise sources in a parallel configuration with input and output noise current sources. Figure 8.42 
shows the transformation of the device noise model with the series thermal voltage noise source 
due to the base resistance into the equivalent device noise model with two parallel current noise 
sources at the input and output only, using the transmission ABCD-parameters. By using formulas 



396 OSCILLATORS 




(a) 




'ni 'nb 



FIGURE 8.42 Transformation of base resistance thermal voltage noise. 



of the transformation from the admittance y-parameters to the transmission AfiCD-parameters, one 
can write 



1 

B 
D 



= -Y 2 l = ~(gm - jcoCc). 
1 



F„ = — + j(v(C K + C c ). 
6 r. 



(8.169) 
(8.170) 



Let us simplify the analytical calculations by taking into account that, as it follows from 
Eqs. (8.169) and (8.170), the contribution of the collector capacitance C c is not significant. As a 
result, the mean-square input current source and output current source Q 0 can be written as 



2qI c Af , 4kTAf /r b y/, , q2 f 2 



il 0 = 2qI c Af + 



AkTAf (I3r b 



(8.171) 
(8.172) 



where/ is the operating frequency,/ T = g m /(2jrC,r) is the transition frequency, and it is assumed that 
the dc and small-signal values of the current gain /3 are equal. 

To calculate the noise figure of the oscillator, the noise current sources (J and should be 
transformed in parallel with i^ R . By using Eqs. (8.166) and (8.167) for the case of ideal junctions 

when n b — n c — 1, the total equivalent noise source i£ s connected in parallel to the resonator and 
load can now be obtained as 



— AkTAf AkTAf 
K,y. = — ^ + 



2/v 



1 + 



2r h 



1 + / 



Ci + C 2 



AkTfSAf / 2M,\ / C 2 \ 2 



(8.173) 



NOISE IN OSCILLATORS 397 



The noise figure F of the oscillator is defined as the ratio of the total noise power due to all noise 
current sources and the noise power from the loaded resonator due to the noise source ;^ R . As a result, 



F= 1 + 



Rl 
2/v 



2n, 



1+^1 + /^ 



C, 



d + C 2 



2r„ V r„ 



C 2 



Ci + C 2 



(8.174) 



In most cases, the contribution of the collector shot noise dominates over the contribution of the 
shot noise caused by the base current. Consequently, for i% r n and/ <^fj, the expression for the 
oscillator noise figure is simplified to 



F = l + -±[l + — 



C, 



Ci + C 7 



(8.175) 



Finally, the single-sideband noise-to-carrier ratio at the output of the Colpitts oscillator in a linear 
consideration, using Eq. (8.161), can be defined by 



L (/ m ) = 



kT 



fo 



2P L \ Ql/„ 



(8.176) 



8.9.4 Impulse Response Model 

The behavior of autonomous second-order weakly nonlinear oscillation systems with low damp- 
ing factor close to linear conservative systems and small time-varying external force /(f) can be 
described by 



dr- 



+ x = n(t) 



(8.177) 



where x is the time-dependent variable, voltage or current, and r = coq t is the time normalized by 
the angular resonant frequency co 0 . 

The phase plane method is one of the theoretical approaches that allows one to analyze qualitatively 
and quantitatively the dynamics of the oscillation systems described by the second-order differential 
equations such as Eq. (8.177) [1]. By setting the small external force equal to zero, the solution of 
the linear second-order differential equation takes the form 



x = Acos (t + (j>) — A cos i/r 
dx 



ill 



= — Asin (r + <p) = — A sin i/r 



(8.178) 
(8.179) 



where A is the amplitude of the oscillations and <f> is the phase of the oscillation. The phase portrait 
shown in Figure 8.43(a) represents the family of circular trajectories enclosing each other with radii 
r = A (limit cycles) depending on energy stored in the system. 

Let us define the variations of the amplitude A(t) and phase <p(t) under the effect of the external 
force applied to the oscillation system [2,49]. Assuming that the effect of the external force is small 
and these variations are slow, the amplitude and phase can be considered constant during a natural 
period of the oscillation. Then, Eq. (8. 177) can be rewritten in the form of two first-order equations by 



dx 
~dx 
dy 



= —x + n (t) . 



(8.180) 
(8.181) 



398 OSCILLATORS 




From Eqs. (8.180) and (8.181) it follows that the instantaneous change of the ordinate y by a 
value of d n y — ndr will occur, due to the small external force injected to the oscillation system. This 
corresponds to a step change of the representative point Mo from the position K to the position L, 
thus resulting in the amplitude and phase changes shown in Figure 8.43(b). These changes can be 
determined from the consideration of a triangle KLN as 

dA — — d n y sinifr — — nsin^/dr (8.182) 

d n y n 
d(p — — — cos \/r — — — cos xjfdt. (8.183) 

Thus, separate first-order differential equations for the time-varying amplitude and phase can be 
obtained from Eqs. (8.182) and (8.183) as 

dA 

— = -nsini/f (8.184) 
dr 



d(p n 

— = cos xlr. 

dr A 



(8.185) 



NOISE IN OSCILLATORS 399 



Since the right-hand sides of Eqs. (8.184) and (8.185) are small, time-averaged differential equa- 
tions can be used instead of the differential equations for the instantaneous values of the amplitude 
and phase. Hence, the changes of the amplitude and phase for a time period t < T are defined as 



AA(t) — — j «(r)sinrdr 
o 

oat 



I — - cos rdr. 



(8.186) 



(8.187) 



Figure 8.44(a) shows the equivalent circuit of a negative resistance oscillator with injected small 
perturbation current i(t). In a steady-state oscillation mode, when the losses in the resonant circuit are 
compensated by the energy inserted into the circuit by the active device, the second-order differential 
equation of the oscillator is written as 



dt 2 



+ (o 0 v 



1 di 
Cdt 



(8.188) 



where co 0 — l/y/LC is the resonant frequency and v(t) is the voltage across the resonant circuit. If 
a current impulse i(t) is injected, the amplitude and phase of the oscillator will have time-dependent 
responses. According to Eqs. (8.186) and (8.187), the resultant amplitude and phase changes have 
quadrature dependence with respect to each other. When an impulse is applied at the peak of the 
voltage across the capacitor, there will be a maximum amplitude deviation with no phase shift, as 
shown in Figure 8.44(£>). On the other hand, if the current impulse is applied at zero crossing, it will 
result in a maximum phase deviation with no amplitude response, as shown in Figure 8.44(c). 

Suppose that a perturbation current i(t), injected into the oscillation circuit, is a periodical function 
that can generally be expanded into a Fourier series 

oo 

i (f) = / O cos Acot + {/kccos [(kcoo + Aa>)t] + / ks sin [(kco 0 + Aco)t]} (8.189) 

k=l 




(a) 




m to 

FIGURE 8.44 Second-order LC oscillator and effect of injected impulse. 



400 OSCILLATORS 



where / 0 is the dc current, / kc is the kth cosine current harmonic amplitude, / ks is the kth sinusoidal 
current harmonic amplitude, and Aco co 0 . In this case, the small external force can be redefined as 

1 di di 

n= = L — . (8.190) 

coqC dr dt 

Consequently, substituting Eq. (8.189) into Eqs. (8.186) and (8.187) and taking into account that 
too + Aco = coo result in 

/i s cos (Acot) — 1 

av(0 = -^ — —r^ — (8- 191 ) 

2C Aco 
/ 1c sin(Acof) 

Am = -2cv^r- (8 - 192) 

where V is the voltage amplitude across the capacitor C. In this case, only the fundamental compo- 
nents in the injected current f(f) can contribute to the amplitude (sine amplitude) and phase (cosine 
amplitude) fluctuations given by Eqs. (8.191) and (8.192), because for the dc and fcth-order cur- 
rent components, the arguments for all their integrals in Eqs. (8.186) and (8.187) are significantly 
attenuated by the averaging over integration period. 

The output voltage of an ideal cosine oscillator with constant amplitude V and phase fluctuations 
A<p can be written as 

v(t) = V cos [a) 0 t + A(f> (t )] = V cos [A(/> (t)] cos co 0 t - V sin [A(f> (t )] sin co 0 t (8.193) 

resulting in an output spectrum of the oscillator with sidebands close to the oscillation frequency 
a>o. Since, for a narrowband phase modulation with small phase fluctuations, sin[A0(f)] = A<f>(t) 
and cos[A0(f)] = 1, the phase modulation spectrum given by Eq. (8.193) can be rewritten using 
Eq. (8.192) as 

u(0 = V cos a) Q t -\ cos [{o^t — Aco) t] cos [{co 0 t + AoS) t] (8. 194) 

4CAa> ACAco 

which is similar to the single-tone amplitude modulation spectrum containing the spectral components 
corresponding to the carrier frequency (o 0 and two close sideband frequencies (o 0 — Am and (o 0 + Aco. 
The injection of the current 

oo 

i{t) — I 0 cos Aa) 0 t + {/kccos [{k(L> 0 — A(o) t] + 4 s sin [{kco 0 — Aco) t]} (8.195) 



has similar effect, resulting in twice the noise power at the sidebands. Therefore, an injected total 
current ;'(?) results in a pair of equal sidebands at a>o ± Aa> with a sideband power P s \, relative to the 
carrier power P c given by 

P s b (&>0 i Aco) 

:2 — . (8.196) 



he V 
:VAco ) 



P c (co 0 ) \4C 

Now let us assume that a stationary thermal noise current with a white power spectral density 
i% is injected into the oscillator circuit close to earner. Then, by making a replacement between the 
amplitude and root-mean-square current values when lf c /2 — i\, the single sideband power spectral 



NOISE IN OSCILLATORS 401 



density for the phase fluctuations at Am offset from the carrier co 0 in \lf 2 region can be written using 
Eq. (8.196) as 

L( ^=4c4a^- (fU9?) 

Finally, taking into account that if = 4FkT/R L for A/ = 1 Hz, Q L = w 0 CR L , and P L = V 2 /2R L , 
where is the tank parallel or load resistance, Eq. (8.197) can be rewritten as 

2FkT ( co 0 \ 2 
L(f m )= — . (8.198) 

which is similar to Eq. (8.161) for the negative resistance oscillator. 

Consideration of both cosine current and cosine voltage across the device output terminals implies 
the device operation in a linear active region only when the establishment of the oscillations can 
be achieved by using a separate diode as a nonlinear element. Therefore, the mixing effect from the 
nonlinear behavior of the active device is not taken into account. For a general case of the active 
device described by a two-port network equivalent circuit, in order to evaluate the output port noise 
voltage generator, it is necessary to provide a transformation of the noise source from the input port 
to the output port of the device. 

As an example, consider an oscillator with a nonlinear output resistance dependent on the applied 
dc bias voltage and on the amplitude of the self-sustained oscillations. The basic oscillator circuit with 
the nonlinear negative output resistance R out , capacitance C, inductance L, load resistance ^l, and 
noise current i n {t) is shown in Figure 8.45(a). The electrical behavior of such an oscillator, in terms 
of voltage v(i) across the capacitance, can be represented by a second-order nonlinear differential 
equation 

d 2 v L dv di 
LC— + —— + v + L-=e ri (t) (8.199) 
dt 2 R h dt dt 

where 

din (0 

e a (t) = L^^- (8.200) 
dt 

is the equivalent noise voltage and 

oo 

' = /o + T~ + £ Gkt,k (8 ' 201) 

Rout k=2 

represents a power series expansion where I 0 is the dc current and are the small coefficients. 



K0 i 




FIGURE 8.45 Second-order nonlinear oscillation systems. 



402 OSCILLATORS 



In a steady-state operation mode, when the active device compensates for the losses in the load 
resistance according to R out + R L — 0, Eq. (8.199) can be rewritten as 

d 2 v d i 

LC dt I + V + L d~t\^ Gk u )= e "W ( 8 ' 202 ) 



k=2 



Seeking the general solution of the inhomogeneous differential equation as the superposition of a 
general solution of the homogeneous (noise-free) and specific solutions of Eq. (8.202) as 

v(t) = V(t)cos [m Q t + 4> (/)] + e n (t) (8.203) 

and applying a van der Pol approach for the slowly time-varying amplitude V(t) and phase cf>(t) allow 
us to rewrite Eq. (8.202) in the form 

dV d(f> 1 d \ 

2a> 0 — sm(a> 0 t + (p) + 2cooV — cos(« o ? + 0)= -— > G k v k (8.204) 
dt dt C dt \ I 

where wo = l/^/LC, It is assumed that e n (t) is a small slowly time- varying low-frequency noise 
voltage, for which 

d 2 e„ (t) 
dt 2 

As a result, substituting Eq. (8.203) into the right-hand side of Eq. (8.204) and using trigonometric 
identities yield 

dd> 

-i-= 0 (8.205) 
dt 

which means that the nonlinear output resistance has no impact on the phase fluctuations. How- 
ever, the amplitude fluctuations are not equal to zero because all factors on the right-hand side of 
Eq. (8.204) have the first-order sine components. Thus, the resistive type of nonlinearities alone would 
cause amplitude noise only, since the reactive elements determining the oscillation frequency remain 
constant. However, if the high-frequency noise current is injected close to the carrier frequency a>o 
(for example, at small offset Acu), it will cause the phase fluctuations according to Eqs. (8.192) 
and (8.204). 

Now consider a varactor-controlled oscillator with the varactor as a nonlinear element, whose 
capacitance depends not only on the applied dc bias voltage, but also on the amplitude of the self- 
sustained oscillations. The basic VCO circuit consists of the varactor with a nonlinear capacitance 
C, an inductance L, and a noise voltage e„(f), as shown in Figure 8.45(Z?) [50]. The voltage e n (t) 
can represent all the noise coming from both inside and outside the circuit, including any thermal 
noise from the resistors, flicker noise from the active device, and noise from the power supply. The 
electrical behavior of the oscillator can be described by 

v + L— =e n (f) (8.206) 
dt 

dC\ dv 

C + v— — (8.207) 
dv I dt 



where the nonlinear term vdC/dv is included in Eq. (8.207). 



NOISE IN OSCILLATORS 



403 



By expanding a nonlinear capacitance C into a power series 



C = c o + J2 Ckvk 



(8.208) 



with the small coefficients C\, substituting Eq. (8.207) into Eq. (8.206) and applying an asymptotic 
perturbation procedure with decomposition of the perturbed and unperturbed equations, the first- 
order differential equation for phase fluctuations with the slowly time- varying noise voltage e n can 
be derived as 



where V is the voltage amplitude across the varactor [51]. Note that a nonlinear capacitance has no 
impact on the amplitude noise of the oscillator. 
From Eq. (8.209) it follows that 

• the first-order capacitance nonlinearity described by the coefficient C\ contributes to the up- 
conversion of the low-frequency noise e n (t) to the sideband noise near carrier «o; 

• the second-order nonlinearity described by the coefficient C2 generates a phase noise, due to 
both amplitude-to-phase conversion and low-frequency noise upconversion; 

• the higher order nonlinearities described by the coefficients Ct, k = 3, 4, 5, . . . , cause a more 
complicated noise behavior of the oscillator based on hybrid upconversion and amplitude-to- 
phase conversion due to the cross-terms of V and e„. 

In the case of a single-frequency LC oscillator, the main contributor to the phase noise is the 
nonlinear collector capacitance of the bipolar device or the gate-source capacitance of the field-effect 
transistor (FET) device. However, in a general case, the equivalent circuit of the active device is very 
complicated, including both nonlinear intrinsic and linear parasitic external elements. This means 
that it is difficult to evaluate analytically the impact of each nonlinear element on the upconversion 
mechanism. Moreover, the joint effect of different nonlinear circuit elements will result in both ampli- 
tude and phase fluctuations. For example, the phase noise can be significantly reduced by linearizing 
both the transconductance g m and the gate-source capacitance C gs , since both nonlinearities are 
important contributors to the phase noise [51]. The amplitude noise also depends on the capacitance 
and transconductance nonlinearities. However, the capacitance nonlinearity will not affect the output 
current if the series gate resistance R g is set to zero. The nonlinearities of the gate-drain capacitance 
Cgd and drain-source resistance R As have negligible effect on the amplitude and phase noise. 

Generally, the transition from soft start-up oscillation conditions to steady-state self-sustained 
oscillations is provided as a result of the degradation of the device transconductance in a large-signal 
mode, when the active device is operated in both pinch-off and active regions. As a result, for the 
cosine voltage across the resonant circuit, the output collector (or drain) current i(t) represents a 
Fourier series expansion 



where I 0 is the dc current and /„ is the amplitude of the nth harmonic component. 

If the oscillation frequency is equal to the resonant circuit frequency, which means that the active 
device has no effect on the oscillation frequency, then the fundamental component of the collector 
voltage will be in phase with the fundamental component of the collector current. However, for all 
higher order voltage harmonics, the impedance of the resonant circuit will be capacitive since the 




(8.209) 



OO 




(8.210) 



404 



OSCILLATORS 



1^ 




v, 



FIGURE 8.46 Schematic of parallel feedback oscillator. 



collector current harmonics are mostly flowing through the shunt capacitance. Therefore, for the 
Meissner oscillator circuit shown in Figure 8.46, the voltage at the input of the active device can be 
written as 



where K ink Vj„i for a high value of the oscillator loaded quality factor. 

As a result, when the device transfer characteristic is approximated by a polynomial, the presence 
of higher order voltage harmonic contributes, first, to the changes in the fundamental amplitude and, 
secondly, to the appearance of the phase shift between the fundamental voltage and the fundamental 
current. In a general form, the frequency deviation Aco caused by the presence of the second-order 
and higher order harmonic components of the voltage v on the resonant circuit can be obtained from 



where k is the order of the harmonic component and m k = VJVi is the ratio of the harmonic voltage 
component to the fundamental voltage amplitude [3]. 

In view of the multiharmonic representation of the oscillator output spectrum due to the device 
and resonant circuit nonlinearities, the total phase fluctuations can be represented by a superposition 
integral as a result of each harmonic contribution. This is similar to the Fourier harmonic expansion 
in a frequency domain of the voltage waveform, when the phase trajectory on the phase plane is a 
result of the phase trajectories with different radii and velocities corresponding to the dc shift and 
harmonic amplitudes. In this case, Eq. (8.187) can take a general form 



oo 




(8.211) 



k=2 




(8.212) 




(8.213) 



0 



where 




(8.214) 



is a dimensionless periodic function characterizing the shape of the limit cycle or phase trajectory 
corresponding to the oscillation waveform and depending on the oscillator topology. It is called the 



NOISE IN OSCILLATORS 405 



impulse sensitivity function (ISF) for an approximate model for the oscillator phase behavior [52] and 
serves a similar role as the perturbation projection vector (PPV) for the exact model [53]. The initial 
phase </>„ in Eq. (8.214) is not important for random noise sources and can be neglected. For an ideal 
case of a purely sinusoidal oscillator, C\ — 1 and F(t) — cosco 0 t. 

Now if any stationary noise current with a white power spectral density i^/Af is injected into 
the oscillator circuit close to any harmonic ncoo + Am or nm 0 — Aw, it will result in a pair of 
equal sidebands at m 0 ± Am. Then, the total single-sideband power spectral density for the phase 
fluctuations in a bandwidth Af — 1 Hz can be written, based on Eq. (8.197), as 



i(/m) 



11=0 

4C 2 V 2 Am 2 



i 2 r 2 



(8.215) 



where r rms is the root-mean-square value of F(t) [54]. Thus, the total noise power near the carrier 
frequency of the oscillator is a result of the upconverted \lf noise near dc, weighted by coefficient 
Co, the noise near the carrier weighted by coefficient C\ , and the downconverted white noise near the 
second-order and higher order harmonics weighted by coefficients c n , n — 2, 3, ... . The converted 
phase noise due to the conversion from one sideband to another can be of the order of 6 dB higher 
than the additive noise in the oscillator [55]. 

From Eq. (8.215) it follows that the effect of the converted phase noise can be reduced by minimiz- 
ing the dc coefficient cq and the higher order harmonic coefficients c„ of the T(?) approximating the 
cosine waveform of the injected node voltage. For a symmetrical flattened drain voltage waveform 
corresponding to a Class F operation mode, the ratio between the voltage fundamental and third 
harmonic should be equal to V1/V3 = 9 with the significantly suppressed even harmonic components. 
On the other hand, for a half-cosine drain voltage waveform corresponding to an inverse Class F 
operation mode, the ratio between the voltage fundamental and second harmonic should be equal 
to V1/V2 — 4 with the significantly suppressed odd harmonic components. Note that, in Class E 
operation with nonsymmetrical voltage waveform, the effect of the second-order and higher order 
harmonics is significant resulting in a high value of the voltage peak factor. The importance of the 
symmetry is necessary also to minimize the coefficient c<j responsible for the low noise upconversion 
and amplitude-to-phase conversion [52], As it is seen from Eq. (8.209), the phase noise improvement 
can be achieved by reducing the effect of the device and circuit nonlinear capacitances. Due to the 
amplitude-to-phase conversion, the phase for each higher order harmonic component changes with 
amplitude resulting in a generally asymmetric voltage waveform. 

Equation (8.213) describes an approximate phase noise behavior, compared with the accurate 
equation where the phase <p(t) also appears in its right-hand side [53]. Such a simplified phase noise 
model is valid for the case of stationary noise sources such as white noise. However, when the noise 
sources are no longer stationary, it can be accurate only in the limits of an assumption of the small 
phase shifts for which cos A(j> is close to unity. This implies that the approximate model is not accurate 
enough to analyze neither the injection-locking phenomenon nor the related issues such as behavior 
of phase differences of coupled oscillators [56], 

The noise sources in an oscillator generally cannot be only modeled as purely stationary since 
the statistical properties of some of them may change with time in a periodic manner. Such types of 
noise sources are referred to as cyclostationary. If the thermal noise of the resistor has a stationary 
nature, then the collector shot noise of the transistor is an example of cyclostationary noise due to the 
time-varying nature of the collector current. The most important issue is that the collector shot noise 
is dominant, compared to the noise from base resistance or tank losses, and can achieve about 70% 
of total phase noise of the oscillator [57]. 

Figure 8.47(a) shows a simplified single-ended common gate complementary metal-oxide semi- 
conductor (CMOS) Colpitts oscillator configuration where the required regeneration factor for the 
start-up oscillation conditions is chosen using a proper ratio of the feedback capacitances Ci and C%. 



406 OSCILLATORS 




The idealized voltage and current waveforms corresponding to the large-signal operation in idealized 
Class B with zero saturation voltage are shown in Figure 8.47(£>). 

Equation (8.210) for the drain time-varying current can be rewritten in the form 



i (0 = Im 



+ ^2 a n cos ( na) o0 



(8.216) 



where a n is the ratio of the nth current harmonic amplitude to the peak output current / max , expressed 
through a half-conduction angle 6 as 



Xn(g) 

1 — cos6> 



(8.217) 



where y n (6) are the current coefficients. 

To account for the cyclostationary drain noise source as a result of total noise sources injected at 
frequencies nco 0 ± Aco, Eq. (8.215) can be rewritten in a general form as 



L (/m) = 



4C 2 V 2 Aco 2 



(8.218) 



where i 2 d — 2ql max is the drain current noise power density in a frequency bandwidth A/ = 1 Hz. 
The Fourier components for the current waveform close to a half-cosine show that the drain shot 
noise is mixed mostly with the fundamental and second harmonics to contribute to the total phase 
noise of the oscillator. 

To minimize the oscillator phase noise, it is very important to choose the optimum value of a 
capacitance feedback ratio k — CilC\ for the same total capacitance C — CyC^iCy + C2). This is 
because different values of the conduction angle correspond to different harmonic contribution to the 
output spectrum. In the case of a Class B with 6 — 90°, the third-order, fifth-order, and higher order 



VOLTAGE-CONTROLLED OSCILLATORS 407 



harmonics can be eliminated since their current coefficients y n (for n = 3, 5, . . .) become equal to 
zero. As a rule-of-thumb, the optimum capacitance feedback ratio for a Colpitts oscillator can be 
chosen to be approximately k — 3.5-4 [52,57]. 



8.10 VOLTAGE-CONTROLLED OSCILLATORS 

The VCOs are key components in many applications, especially in wireless communication systems, 
measurement equipment, or military applications. A growing market of wireless applications requires 
highly integrated circuit solutions, where both high-performance transistors and passive elements with 
high-quality factors can be used. To analyze the tuning linearity of the VCO circuit, a general approach 
describing the oscillation circuit in terms of natural frequencies of a lossless two-port network when 
one of its ports is short circuited can be used [58]. Figure 8.48 shows the block diagram of the general 
VCO equivalent circuit, where a linear lossless network can incorporate one or several resonant 
circuits including active device reactive elements, the baseband modulation signal is brought to the 
varactor using port 3-3', and the load is connected to the port 4^1'. Such a block representation 
enables the description of the VCO modulated curve in terms of poles and zeros, irrespective of any 
particular circuit diagram. 

The oscillation frequency can be found form the phase balance condition of 



(8.219) 



where X^coo) is the reactance seen by the varactor, X v (coq) is the reactance of the varactor, and coo — 
27r/ 0 is the radian oscillation frequency. The reactance seen by the active device is equal to zero at 
the oscillation frequency. 

Generally, the reactance X in can be expressed in terms of poles and zeros. For instance, for the 
case shown in Figure 8.49 where the poles occur at the origin and co 2 while the zeros occur at a>i and 
0)3, the reactance X m is written as 



X m (u) = K 



(&j 2 — to 2 ) (ft) 2 — ft) 2 ) 
ft) (ft) 2 — of) 



(8.220) 



The reactance due to the varactor junction capacitance can be written using Eq. (2.6) in Chapter 2 
in the form 



X v = 



a)C V0 <pr 



(8.221) 



where the voltage u v on the varactor consists of the dc bias voltage V v and of the modulation voltage v m . 
The modulation voltage can be represented in the normalized form 



V Y + <p 



(8.222) 



4 

4' 







Linear 


< 


Active 
device 








lossless 
ii eL work 


c 









Varactor 



3 

.V 



FIGURE 8.48 Block diagram of VCO with linear circuit. 



408 OSCILLATORS 




FIGURE 8.49 Reactance of lossless network. 



For the initial conditions of a> = co 0 , v is equal to zero. This leads to 



(w 2 - oj 2 ) (oj 2 - oj 2 ) (co 2 - co 2 ) 
Introducing the frequency normalization in the form of 



(8.223) 



co 0 



= £2, fori = 1,2, 3 



and taking into account that, in the vicinity of the operating point 

1 + rj 



CO 

eo 0 



(8.224) 



(8.225) 



where ij — (co — co 0 )/oj 0 is the normalized frequency change representing a very small number, the 
modulation curve can be rewritten by 



1 + v = 



(1 + i]f - Q\ 1 - Q 2 2 (1 + rff - Q,\ 
(1 + ,;) 2 - Q.\ l-Q 2 3 



(8.226) 



When the linear lossless circuit between the varactor and the active device has more than three 
finite natural frequencies, Eq. (8.226) can be appropriately expanded to include any additional zeros 
and poles. 

The simplest two-port network between the varactor and the active device is a single series resonant 
circuit shown in Figure 8.50(a)- The modulation curve for such a resonant circuit can be obtained 
from Eq. (8.226) as 



l + v = 



1 + Air] + —rf 



i/r 



(8.227) 



VOLTAGE-CONTROLLED OSCILLATORS 409 



L, 



-o a 
2' I 



I f.', .W C f 2 , C, t, 



— Q O- 

2' r 



(/>) •=■ 

FIGURE 8.50 Schematics of two-port networks. 



where 



1 - Qf 



Since 17 is a small number, it is convenient to use the binomial expansion to obtain first two terms 
of the Taylor series for voltage v by 



where 



C, = 



C 2 = 



V = C\T] + C2f) 



A, 



(8.228) 



Y 

Ai 

2y 



1 + 



1 A, 



The slope coefficient Ci is the inverse of the normalized circuit sensitivity 5„ equal to 



c _ dr > 
dv 



/o 



(8.229) 



where S is the slope of the actual circuit sensitivity (or modulation curve) in hertz per volt. As a 
result, for the series resonant circuit shown in Figure 8.50(a), the normalized sensitivity must satisfy 
the condition of 



(l-n?). 



(8.230) 



The modulation curve is linearized by requesting C2 = 0 resulting in 

2 

Y 



(8.231) 



From Eq. (8.231) it follows that the varactor junction sensitivity y must be larger than unity 
because Q { < 1. Consequently, the only possibility to linearize the modulation curve for a series 
resonant circuit is to use a hyperabrupt varactor. However, such a resonant circuit for a fixed operating 
point does not provide enough flexibility to accommodate small variations in y . 

The inductively coupled pair of lumped resonant circuits shown in Figure 8.50(b) offers more 
degree of freedom to linearize the modulation curve, having the position of natural frequencies o)\, 
a)2, and (03 (when port 2-2' is short-circuited), as shown in Figure 8.49. To achieve resonance with 



410 OSCILLATORS 



the varactor capacitance, reactance X in must be positive with oscillation frequency a> 0 located above 
the zero 0)3. 

The linearity of the modulation curve of the resonant circuit with two coupled resonators can be 
defined by expanding Eq. (8.226) in a three-term Taylor series 



v - C\T) + C 2 r) + C 3 ri 



(8.232) 



where 



d = — (Ai — A 2 + A3) 

y 



c 2 



Ci + C\ - - {A\ - A\ + A3) 



C l C l Ci 
"1 2 ~ T 



+ C 2 Q + + ^-{A\-A\ + A3) 



where 



Aj 



1 - S2? 



- for; = 1,2,3. 



Since the normalized sensitivity 5„o at the center frequency co 0 can be written as S n o — 1/Ci, and 
the linearity of the modulation curve requires C 2 — C3 = 0, then the three nonlinear equations can be 
obtained as 



A 2 + A 3 



y_ 

Sua 



(8.233) 



Ai 



At "I - A3 



Ao + A3 — 



y_ 

St® 



1 + 



'nl) 



3 1 /3 1 

2 5 n o \2 S„o 



(8.234) 
(8.235) 



which should be solved numerically to find the optimum position of natural frequencies forcing 
elimination of both the second and third coefficients in Eq. (8.232). 

The calculated values Qi, Q 2 , and ^3 are substituted in Eq. (8.226), and the relative frequency r) 
is gradually varied so that normalized modulation curve v{r\) is evaluated [58]. The higher the chosen 
value of S n o, the wider bandwidth within which the deviation from linearity stays within prescribed 
limits. For example, by using an abrupt varactor with y — 0.5, for 1% deviation from linearity, the 
sensitivity 5„o = 0.02 results in the relative bandwidth of 0.0214, while the relative bandwidth is 
0.0319 for S„o = 0.03. To provide oscillation stability, the following approximate condition should 
be satisfied: 



ko y n r 



(8.236) 



where £2 r = a> t /coo is the normalized resonant frequency associated with the varactor side, 



C 2 + C v 
L2C2 C\ 



VOLTAGE-CONTROLLED OSCILLATORS 411 



<T2 V = a> v /(o 0 is the normalized resonant frequency associated with the varactor only, 



1 



k 2 — M 2 /LiL 2 is the inductive coupling coefficient, and Q v is the varactor quality factor. 

To increase the frequency tuning bandwidth of a negative resistance oscillator, a reactance com- 
pensation technique based on the tandem connection of a series resonant circuit and parallel resonant 
circuit both tuned to the fundamental frequency, as shown Figure 8.50(c), can be used [2,59]. Such a 
reactance compensation technique can also provide a linearization of the frequency tuning character- 
istic for the certain ratios between the oscillator circuit parameters [60]. For example, a linear tuning 
bandwidth of 1.7 GHz with a minimum slop ratio of 1.08 can be achieved for an X-band negative 
resistance oscillator. 

Depending on the operating frequency and application requirements, there is a variety of VCO 
implementation techniques based on using different types of the active devices, circuit schematic 
approaches, and hybrid or monolithic integrated circuit technologies. For example, their low cost 
implementation and low-phase noise performance are required in wireless communication systems. 
At microwaves, most of these VCOs use MESFET or heterojunction bipolar transistor (HBT) devices 
because of producing lower phase noise than VCOs based on high electron mobility transistor (HEMT) 
devices. However, high performance HEMT oscillators are essential if they can be integrated together 
with amplifiers and mixers for single-chip receivers or transmitters using the same technology. 

The common gate VCO configuration is usually used to generate strong negative resistance over 
a wide frequency range. However, due to the series resistance of the varactor, generally the phase 
noise of such a VCO is relatively higher than that of a single-frequency oscillator. Nevertheless, the 
phase noise can be reduced by placing the varactor into the source rather than into the gate circuit. 
Figure 8.51 shows the circuit schematic of a common gate MESFET VCO MMIC where an output 
L 2 Ci matching network is incorporated in the form of a high-pass section to eliminate low-frequency 
parasitic oscillations [61]. Based on an analysis of the trajectories of the reflection coefficient lines of 
the resonator r in and the device r out as a function of the drain supply voltage, the oscillator loaded 
quality factor Q L was maximized. As the drain voltage increases, the angle between these trajectories 
approaches 90° where <2l becomes maximal [62] . Increasing the drain voltage also reduces the phase 
noise by extending the depletion region in the device channel to the drain side, thus reducing the 
sensitivity of the oscillator to the gate-source voltage. Being implemented in a commercial 0.6- p.m 
GaAs MESFET process, such a VCO demonstrates the phase noise of —91 dBc/Hz at 100 kHz offset 
with an output power of 1 1.5 dBm and frequency tuning of 500 MHz around the center frequency of 



Wideband VCOs are used in a variety of RF and microwave systems, including broadband 
measurement equipment, wireless and TV applications and military electronic countermeasures 



11.5 GHz. 



Vy O 




o V M 



-o 

Output 



FIGURE 8.51 Circuit schematic of common gate MESFET VCO. 



412 OSCILLATORS 





0 2 4 6 8 
(h) 

FIGURE 8.52 Equivalent circuit of common collector bipolar VCO. 



(ECM) systems. In modern ECM systems, they serve as the frequency-agile local oscillators in receiver 
subsystems and fast-modulation noise sources in active jamming subsystems. Among wideband 
tunable signal sources such as YIG-tuned (yttrium iron garnet) oscillators, wideband VCOs are 
preferable because of their small size, low weight, high settling time speed and capability of fully 
monolithic integration. Therefore, modern radar and communication applications demand VCOs that 
are capable of being swept across a wide range of potential threat frequencies with a speed and 
settling time far beyond of the YIG-tuned oscillators. 

The tuning possibility of a bipolar VCO resonant circuit can be evaluated by using the simple 
device equivalent circuit shown in Figure 8.52(a) [63]. Here, the collector terminal is common and 
usually RF grounded in the practical realization with a bypass capacitor. The simplified equivalent 
circuit includes the collector capacitance C c , the base-emitter capacitance C % (including diffusion 
and junction capacitances), and a current source described by the small-signal transconductance g m . 
To provide wideband frequency tuning, the two varactors and are included into the base and 
emitter circuits, respectively. 

For such a common collector bipolar VCO, the equation for resonant frequencies in a steady-state 
operation mode can be given by 



m z LC b v C c + 



C n + C[ 



c° + c c + 



c„c v e 



(8.237) 



To characterize the VCO band properties, it is convenient to use the generalized dependencies 
Kf(K cl , K c2 ), where K cl = C^/C^ and K c2 = C' max /C' lllin , and the normalized parameters to 
obtain the results regardless of the particular values of the circuit parameters. By using the normalized 



VOLTAGE-CONTROLLED OSCILLATORS 413 



parameters m 0 = ai T C c /g m , qi = Cc/Cj^, and q 2 = Cc/C^, Eq. (8.237) can be rewritten in a 
general form 

K L (1 + qi)(m Q + q 1 ) + q l K c2 (l+m 0 ) + q 2 (8 238) 

f V Ci (qi + K cl )(m 0 K c2 + q 2 ) + qiK c2 1 + m 0 + q 2 

Figure 8.52(i) shows the different dependencies K[(K cl , K c2 ) for various values of qi and q 2 and 
fixed value mo — 0.012. Here, curve 1 is plotted for qi — 1 and q 2 — 0.5 with varactor tuning in 
the base and emitter circuits simultaneously. Curve 2 is characterized by 51 = 1 and q 2 — 0.05 with 
varactor tuning only in the base circuit when K c2 — 1. Curve 3 is calculated for qi — 0.1 and q 2 — 
0.5 with only varactor tuning in the emitter circuit when K c i — 1. A comparison of the curves shows 
that, for varactor tuning in the base and emitter circuits simultaneously, maximum tuning bandwidth 
is achieved (curve 1). Using varactors only in the base circuit (curve 2) gives larger tuning bandwidth 
than in the case of the only varactor tuning in the emitter circuit (curve 3). In this case, decreasing 
q 2 and increasing q\ can increase the tuning bandwidth. To increase the tuning bandwidth only by 
emitter varactor tuning, it is necessary to reduce the parameter q\ significantly, provided q 2 — 1. 

Figure 8.53 shows a typical common collector lumped VCO circuit where the two back-to-back 
varactors provide a wideband tuning [19]. In this circuit, the load is conductively connected to the 
resonant circuit inductor. The choke inductors L ch and bypass capacitors Cb form the low-pass filters 
having high impedance at the fundamental frequency to isolate voltage supplies and low impedance 
for the modulation frequencies. By using abrupt varactors with K c — 3 in a bias voltage range from 
0 to 5 V and minimum capacitance C vm j n = 0.6 pF, it is possible to provide a wideband tuning 
in a frequency range with Kf — 1.6. Similar tuning bandwidth is predicted by curve 2 shown in 
Figure 8.52. Taking into account that the equivalent device input capacitance is equal to 1.5 pF, the 
tuning bandwidth from 5.5 to 8.0 GHz was achieved with a tank inductor of L — 1 .9 nH. 

However, for a common collector VCO, the conductive load connection is not the only way to 
obtain a maximum level of output power. In addition, it is very important to provide its minimum 
flatness over the entire tuning bandwidth. The load can also be connected to the emitter terminal, thus 
decreasing the influence of the load impedance on the resonant circuit that enables one to provide its 
higher quality factor. Figure 8.54 shows (a) the simplified common collector VCO schematic and (b) 
two possible combinations of the admittances in the base and emitter circuits. In the first case, the 
load is connected to the resonant circuit conductively or inductively, provided the impedance in the 




FIGURE 8.53 



414 OSCILLATORS 



I 



I I 









1 






•> 







(a) (*) 
FIGURE 8.54 Simplified VCO schematic with two combinations of base and emitter circuits. 



emitter circuit is capacitive. The second combination requires inductive impedance in the base circuit 
when the load is connected in parallel to the device emitter and collector terminals. 

Figure 8.55 shows the calculated dependencies of the normalized output power versus normalized 
frequency for both above-mentioned cases [64]. For case 1, the load is connected to the base circuit 
and the output power has maximum when 0) = co a , where co a — 2jrf a ,f a is the alpha cutoff frequency. 
For case 2, the output power comes from the device emitter and its level changes negligibly up to 
w = 0.5 a> a . As a result, for the latter case, VCO can be tuned easily in a very wide frequency range 
by a simple tuning of the value of the series inductance in the base circuit using a varactor diode in 
reverse bias operation. The series or parallel i?C-circuit with constant capacitance and load resistance 
can provide the capacitive impedance in the emitter circuit. 

Figure 8.56 shows the electrical circuit of a monolithic common drain MESFET VCO designed 
for a phase-locked loop application in the telecommunication system operating at 14 GHz [65]. A 
planar Schottky-barrier diode with 0.5 x 280 p.m 2 stripe suitable for monolithic integration with the 
MESFET device of the same gate geometry is used as a varactor. The varactor junction capacitance is 
tuned from 1 .0 to 0.5 pF by applying a reverse bias voltage from 0 to 5 V. To provide inductive reactance 
in the gate circuit, the microstrip line is used as a quarterwave transformer between the varactor diode 
and the MESFET device. A 50- £2 resistor was inserted into the source circuit for self-biasing resulting 




VOLTAGE-CONTROLLED OSCILLATORS 415 



0 




FIGURE 8.56 Schematics of microwave MESFET VCO. 



in a drain current stabilization. Another 50-£2 resistor was connected to the microstrip impedance trans- 
former to provide a gate zero bias point. These resistors are necessary to prevent low-frequency oscilla- 
tions by contributing to the significant decrease of the loaded quality factor at frequencies much lower 
or higher than resonant frequency. An output series capacitance is small enough to reduce the pulling 
effect, compromising the output power. The frequency tuning bandwidth from 11.3 to 14.3 GHz 
was achieved by varactor tuning in a bias range from 0 to 7 V with an output power of —4 dBm. 

Figure 8.57 shows the wideband VCO circuit that was used to design and fabricate four GaAs 
monolithic VCO chips that cover 2-4 GHz, 4-7 GHz, 7-12 GHz, and 12-18 GHz frequency ranges, 
respectively [66]. Each monolithic chip includes the MESFET device, two varactors, gate inductor L g , 
source inductor L s , and two bypass capacitors Cb- The varactor diode represents a single implanted 
structure into semi-insulating material and is formed with the same active region as for the MESFET 
device. Its capacitance ratio is typically of 8:1 or greater. The common gate VCO configuration 
using the gate inductor as a regenerative feedback element exhibits negative impedance at the source 
terminal across the entire tuning bandwidth for the equivalent load resistance connected to the drain 




FIGURE 8.57 Microwave monolithic wideband VCO circuit schematic. 



416 OSCILLATORS 



terminal of approximately 15 Q. To provide the output matching with standard load of 50 £2, the 
L-type matching circuit with a series inductor L { and a shunt capacitor Ci is used. Due to the required 
additional area, the output matching circuit was not included on the chip. 

The VCO performance in terms of phase noise and tuning range determines the basic characteristics 
of a whole transceiver. However, a limited frequency-tuning range is usually a serious problem for 
VCO fully based on CMOS technology. There are four basic candidate differential topologies shown 
in Figure 8.58 [67]. Figure 8.58(a) shows an nMOS differential VCO topology with tail current 




— (O (</) 
FIGURE 8.58 Single differential topologies of CMOS VCO. 



CRYSTAL OSCILLATORS 417 



source, while Figure 8.58(b) shows an nMOS differential topology with top current source. The 
pMOS differential topologies with tail and top current sources are shown Figures 8.58(c) and 8.58(d), 
respectively. As regards tuning range capability, topologies shown in Figures 8.58(b) and 8.58(c) are 
preferable. This is because the anode varactor bias voltage is fixed to V dd for an nMOS oscillator with 
tail current source, and the anode varactor bias voltage is fixed to zero for a pMOS oscillator with top 
current source. 

Since using pMOS devices results in a lower phase noise due to inherently smaller low-frequency 
\lf noise, the pMOS differential topology with tail current source shown in Figure 8.58(c) represents 
the best choice for a low-noise wideband tuning. Being implemented in a 0.25-iim standard CMOS 
process, such a differential VCO with a Q-factor of the tank inductor of 7.5 provides the phase noise 
of — 109 and — 123 dBc/Hz at 100 and 500 kHz offset from earner frequency of 1.3 GHz, respectively. 
The tuning range was 13.3% for V di = 1.4 V and 20.1% for V M = 2.0 V. 

Using a proper configuration of the accumulation MOS varactors can significantly improve the 
tuning range. Such a varactor with high quality factor fabricated using the 0.13-|a.m CMOS silicon- 
on-insulator (SOI) technology demonstrates the capacitance ratio of 5 with ± 1 V voltage tuning 
that provides over 50% frequency-tuning range [68], However, a high capacitance ratio implies a 
high varactor sensitivity or ATyco, which makes the oscillator phase noise worse. A band switching 
solution can reduce the varactor sensitivity, but requires extra control circuitry. A simple and effective 
solution can be provided by differential varactor tuning to avoid the effect of high A" V co- 

8.11 CRYSTAL OSCILLATORS 

The traditional and most common type of a piezoelectric resonator used in electronics is the quartz 
crystal. Its operation is based on a piezoelectric effect that converts the electrical signal applying to 
the opposite sides of the crystal to mechanical motion and reconverts the vibratory motion of the 
crystal back into an electrical signal at the resonator terminals. The amount of motion varies over wide 
extremes depending on how closely the applied signal frequency approaches a natural mechanical 
resonance of the crystal. In a properly designed resonator these regions of high-amplitude mechanical 
vibration are very narrow in frequency and are ideally suited for oscillator stabilization. Therefore, the 
oscillator circuits using a quartz crystal are called the crystal oscillators. The first crystal-controlled 
oscillator, using a crystal of Rochelle salt, was first built by Alexander M. Nicolson and the results 
were filed in the form of a patent application in 1918 [69]. In 1919, Walter G. Cady used the quartz 
to control the frequency of an oscillator and described the use of quartz bars and plates as frequency 
standards and wave filters [70]. Since then, it is generally accepted that Cady was the first to use the 
quartz to control the frequency of an oscillator circuit. However, subsequent litigation resulted in a 
legal decision in favor of Nicolson who is therefore considered to be the inventor of the piezoelectric 
oscillator. 

The quartz crystal represents one of forms of silicone dioxide (Si02) that is found in nature and is 
ideal for use as a frequency-determining device because of its predictable thermal, mechanical, and 
electrical characteristics with a high quality factor. The quartz properties are primarily determined by 
the orientation angle at which the quartz wafers are cut, being dependent on the reference directions 
within the crystal that are referred to as axes. There are three axes in quartz, forwarded along the X, Y, 
and Z directions. Ideal quartz would consist of a hexagonal prism with six facets at each end, with a 
cross section shown in Figure 8.59 [71]. The Z-axis is known as the optical axis, repeating its physical 
properties every 120° as the crystal is rotated about the Z-axis. This axis is not anisotropic to light; 
therefore light passes readily. The X-axis called the electrical axis is parallel to a line bisecting the 
angles between adjacent prism faces. Electrical polarization occurs in this direction when mechanical 
pressure is applied. The F-axis, which is called the mechanical axis, runs at right angles through the 
face of the prism, and at right angles to the X-axis. 

The AT cut is the most popular of the Y-axis group because of its excellent temperature character- 
istic, and it is produced by cutting the quartz bar at an angle of approximately 35° 15' from Z-axis. The 



418 OSCILLATORS 



I 




I 



FIGURE 8.59 Typical crystal cuts from a doubly terminated quartz crystal. 

crystal resonator is usually represents a round disc and its fundamental mode measured in megahertz 
is defined as 

N 

f=-r (8.239) 
a 

where d is the disc thickness in millimeters and A' is the frequency constant for particular cut, for 
example, equal to 1.661 MHz/mm for AT cut, 1.797 MHz/mm for SC cut, or 2.536 MHz/mm for 
BT cut [72]. Because the frequency of the crystal is related to its thickness, there is a limitation in 
the manufacturing of high-frequency fundamental crystals. For example, a 100 MHz AT-cut crystal 
resonator has a thickness of only 16.61 microns. In this case, the higher the frequency, the thinner 
should be the crystal plate. Such a thin wafer is not only somewhat fragile but also impractical 
to fabricate by conventional means, which are normally limited to thicknesses of 30-35 microns 
(approximately 50 MHz for AT-cut resonators). However, by introducing chemical polishing methods 
using fluorides and composite double inverted mesa blank structures with air-gapped electrode have 
resulted in AT-cut crystal resonators with fundamental frequencies up to 1.6 GHz, corresponding to 
a resonator plate thickness of just less than 1 micron [73,74]. The measured quality factors of these 
units ranged from 73,000 at 100 MHz and 32,000 at 250 MHz to 5,000 at 950 MHz. 



CRYSTAL OSCILLATORS 419 



,o 0 or 



Co 




FIGURE 8.60 Simplified equivalent circuit of crystal resonator. 



Figure 8.60(a) shows the symbol for a crystal resonator that should generally be treated as a 
two-port device together with its metallic enclosure, leads and supports. However, in some practical 
cases, especially at lower frequencies, the complete equivalent circuit of a crystal resonator can 
be simplified, as shown in Figure 8.60(7:?), where the crystal vibration portion is represented in the 
vicinity of a series resonance by dynamic parameters such as an inductance L\, a capacitance C\ 
and a resistance R\, whereas Co is the static capacitance associated with the crystal and its adherent 
electrodes plus the stray capacitances internal to the crystal enclosure [74]. The impedance of a crystal 
resonator (between point 1 — 1') can be written as 

2 , ■ 2 
j cof + jco~-co 

2 (jco) = — |i . (8.240) 

<o 2 + jay- ~ o> 



where 



co, = , 1 (8.241) 



is the series resonant frequency in radians per second and, for Co 2> C\ , 



is the parallel resonant frequency in radians per second. 

The simplified model of a crystal resonator shown in Figure 8.60(£>) can be represented by an 
equivalent general network, as shown in Figure 8.60(c), with the impedance expressed as 

Z(jco) = R c {co) + jX t (co) (8.243) 

where 



gi 

(1 -abf + b 2 



R e (<») = V, , 2 . (8.244) 



(1 - abf + b 2 
where a — (co s Li/Ri)(co/co s — cojco) and b — coCqRi 



a (1 — ab) — b 

X e (co) = Ri — / (8.245) 



420 OSCILLATORS 



R,. X e 




FIGURE 8.61 Resistive and reactive parts of crystal resonator network. 



The equivalent-circuit resistance R e and reactance X e given by Eqs. (8.244) and (8.245), re- 
spectively, are plotted in Figure 8.61 as functions of frequency over its wide range. It is seen that 
reactance X e is zero at two values of frequency where the lower frequency is just slightly above the 
series resonant frequency co s and upper frequency corresponds to a nearly maximum resistance R e . 
Equation (8.245) can be simplified to 

2 Af 

X e (a>)«— -f (8.246) 

which is used as a measure of the slope of the curve at the operating frequency /. The slope of X e 
versus frequency is a measure of crystal resonator stiffness [72], The resonator quality factor Q x is 
defined as 

2x = — l — (8.247) 
coCiRi 

the inverse value of which is a measure of loss in the resonator. Therefore, the resonator (or unloaded) 
quality factor is always higher than the oscillator (or loaded) quality factor. 

Generally, the crystal oscillators can be divided into two basic groups. In the first group, the 
crystal resonator represents an inductive reactance, whereas, in the second group, it is used as a series 
resonant circuit included into the feedback loop. The basic and most widely used crystal oscillator 
circuit corresponding to the first group is shown in Figure 8.62(a), where the crystal resonator is 
included between collector and base terminals. It is called the Pierce oscillator in the honor of 
George W. Pierce, whose modified and simplified single-stage crystal oscillator circuit became the 
touchstone of the radio communication art. It is also possible to include crystal oscillator between 
the base and emitter terminals, as shown in Figure 8.62(b), and between the collector and emitter 
terminals, as shown in Figure 8.62(c). In both latter cases, the crystal oscillator represents an inductive 
three-point circuit, which is a crystal analog of a Hartley LC oscillator, unlike of a Colpitts LC analog 



CRYSTAL OSCILLATORS 421 






(«) </» (c) 

FIGURE 8.62 Simplified oscillator circuits with crystal resonator. 



of a crystal Pierce oscillator. Adding additional capacitance across the crystal will cause the parallel 
resonance to shift downward. This can be used to adjust the frequency at which a crystal oscillator 
oscillates. 

For an ideal Pierce oscillator shown in Figure 8.62(a) when the active device is regarded as an 
ideal and loss in crystal is neglected, the phase balance condition can be written as 



Xl (ft>osc) + X 2 (« osc ) + X e (&> OS c) = 0 



(8.248) 



where a) osc is the oscillation frequency, X\ — l/(«; 0SC Ci), X 2 — 1/(&> 0 scC2), andX e (&> 0 sc) is defined by 
Eq. (8.245). The graphical solution of Eq. (8.248) is shown in Figure 8.61, where there are two points, 
in which a curve —(Xi + X 2 ) intersects a curve X e . In this case, point A corresponds to the oscillation 
frequency a> osc with a low resistance R e , whereas point B corresponds to the frequency with a high 
resistance R e where the amplitude balance condition is not satisfied and build-up of self-oscillations 
can never happen. An analytical solution of Eq. (8.248) determines the frequency of self-oscillation 
equal to 



1 + 



1 



2 Co + C 2 C 3 / (C 2 + C) 



(8.249) 



Most types of the quartz vibration may be used on different overtones or harmonics. Among them, 
the most popular by far are the thickness vibrations of plates such as AT- and BT-cut of quartz, where 
the thickness determines the fundamental frequency according to Eq. (8.239). Odd overtones of the 
fundamental frequency may be excited in resonator, and Figure 8.63(a) represents the equivalent 
circuit of a family of overtones associated with the fundamental, where Cn = Ci/N 2 , Ln = Li, and 
i? N — RiN 2 , N — 1, 3, 5, ... is the overtone number [72]. It should be noted that overtones are 
used because the smaller Cn values lead to greater frequency stability due to increasing reactance 
slope with increasing N. In order to operate at one of the crystal overtones, the oscillator schematic 
should be slightly changed. In this case, an additional inductor L can be connected in parallel to a 
capacitor C\, as shown in Figure 8.63(£>). The resonant frequency of LC\ circuit is chosen to be lower 
than the required operating frequency but higher than the nearest overtone. As a result, this circuit 
represents a capacitive reactance at the operating frequency, providing a conventional capacitive three- 
point oscillator configuration. However, at lower frequencies, the LC\ circuit represents an inductive 
reactance when the phase balance condition is not satisfied. When the crystal is connected as the 
transistor emitter load impedance, such a harmonic emitter-coupled crystal oscillator can provide 
very good short-term frequency stability [75,76]. 

To design crystal oscillator at higher frequencies, it is necessary to maximize its regeneration 
factor, which can be increased by minimizing the effect of the internal transistor feedback collector- 
base capacitance, thus resulting in a higher device maximum oscillation frequency. This can be 
done by using a cascode crystal oscillator configuration, simplified schematic of which is shown in 



422 OSCILLATORS 



9 9 

Co 




i I 
i I 




(a) 

FIGURE 8.63 Crystal oscillator overtone operation. 

Figure 8.64(a) [77]. In this case, the small-value collector-emitter capacitance of a common base 
transistor is connected in series with the high-value collector-base capacitance of a common emitter 
transistor. As a result, the operating frequencies of a crystal oscillator using the fundamental quartz 
or SAW resonator can be extended up to 1 GHz by using the same transistors. 

In addition to the oscillator configuration where the crystal resonator operates as an inductance, 
there is a popular member of a family of the crystal oscillators where the crystal resonator is used 
as a series resonant circuit. Such a crystal oscillator, the circuit configuration of which is shown 
in Figure 8.64(£>), is called the Butler or Bridged-T oscillator [72]. An operation principle of this 
oscillator is based on the fact that the real part of the crystal impedance at its series resonant 
frequency co s is minimal and drastically increases at the frequencies far from a> s . Therefore, if a 
crystal resonator is included into the feedback loop, for example, between the external LC resonant 
circuit and the transistor emitter, as shown in Figure 8.64(b), then, at frequencies close to the series 
resonant frequency co s , the feedback loop becomes a closed loop and soft build-up of the self- 
oscillations occurs. At the same time, at frequencies different from a> s , the real part of the series 
crystal impedance is high, resulting in a small regeneration factor, and the amplitude oscillation 
condition is not satisfied. Such an operating mode has the advantage that its amplitude is optimized 
since the regeneration factor is maximized. However, the frequency stability is not at maximum level 
in this case, and to maximize frequency stability, it is necessary to include an additional reactive 
element (capacitance or inductance) in series to the crystal resonator, depending on the character of 




(a) 



ri 














HDh- 








= c 2 



(h) 



FIGURE 8.64 Schematic of cascode and Butler crystal oscillators. 



DIELECTRIC RESONATOR OSCILLATORS 



423 



Resonator 
network 



Terminating 
network 



(a) 



3.5 pF 




(*) 



50O 



FIGURE 8.65 Schematics of negative resistance crystal oscillator. 



the loading reactance of the external resonant circuit. It should be noted that the parasitic oscillations 
can be arisen at frequencies significantly higher than a> s due to the crystal static capacitance Co, whose 
reactance is small at these high frequencies and can result in a significant feedback with potential 
establishing of self-oscillations. To prevent this parasitic phenomenon, it is necessary to include an 
external neutralizing inductance Lo in parallel to Co, in order to form the parallel resonant circuit at 
very high frequencies. 

The negative resistance approach to design a crystal oscillator can also be attractive, based on 
a common base transistor configuration with a series feedback inductor at the base, as shown in 
Figure 8.65(a) [78]. In this case, the parasitic capacitances of the transistor, together with the series 
inductive feedback obtained by a short-circuited stub, and a properly designed terminating network 
provides the negative resistance required at the input port, connected to the resonant network, to satisfy 
the oscillation conditions in a wide frequency range. Figure 8.65(6) shows the complete schematic 
of a microwave crystal oscillator operating at the fundamental resonance of the crystal resonator of 
842.911 MHz. The oscillator was built using a substrate with relative dielectric constant of 4.8 and 
thickness of 1.5 mm. The experimental results showed that the negative resistance approach to design 
the crystal oscillator is well suited for its potential low-noise microwave application. 



8.12 DIELECTRIC RESONATOR OSCILLATORS 

It is known yet from 1930s that the dielectric object with free-space boundaries can resonate in 
various modes [79]. If the relative dielectric constant is high, the electric and magnetic fields of 
a given resonant mode will be confined in and near the resonator and will attenuate to negligible 
values at a distance small compared to free-space wavelength. Therefore, radiation loss is very small, 



424 OSCILLATORS 



and the unloaded Q of the resonance is limited mainly by losses inside the dielectric body. Since 
the magnetic permeability of the dielectric material used is unity, then magnetic losses are zero. 
Electrical losses occur as a result of the finite loss tangent (tan<5) of the dielectric material. Typical 
taniS values for dielectric materials used as resonators are about 0.0001-0.0002, thus resulting in 
the values of unloaded Q of about 5000-10,000 and even higher for high-purity TiCK disks [80]. 
For the fundamental-mode resonance, the dimensions of a dielectric resonator are on the order of 
one wavelength in the dielectric material. As a result, for the material with high relative dielectric 
constant, the size of the dielectric resonator can be very small. For example, for a polycrystalline TiC>2 
ceramic with a relative dielectric constant e r of about 100, the effective wavelength in the dielectric 
will be one-tenth of the wavelength in air only. However, the disadvantage of dielectric resonators is 
significant variation of dielectric constant with temperature. For Ti02 material, the relative change of 
dielectric constant e I is about 1000 ppm/°C. Higher dielectric -constant materials such as strontium 
titanate (s r > 250) have much greater temperature sensitivities. Therefore, materials with dielectric 
constants 10 £ e r S 100 are generally used, such as barium tetratitanate and titanium dioxide. 

The most practical shape of a dielectric resonator is a cylindrical disk whose length (thickness) 
L is less than its diameter D. With this shape, the lowest frequency resonant mode has a circular 
electric field distribution, whereas the magnetic field is strongest on the axis of the disk and at a 
sufficient distance outside the disk the field resembles that of an axial magnetic dipole, as shown in 
Figure 8.66 [81]. The axial-dipole mode offers the best separation from other resonances when the 
dielectric optimum ratio of LID is about 0.4 [80]. The resonant frequency of the dielectric resonator 
depends on its shape, being a function of the relative dielectric constant e r , diameter D, and thickness 
L for a cylindrical resonator configuration [82]. 

In practical microwave resonant circuits, the dielectric resonator is placed on the alumina substrate 
and is confined within a metal case. In this case, the resonant circuit has a tripled-layered structure of 
alumina substrate, cylindrical dielectric resonator, and air gap. As a result, the resonant frequencies 
of such tripled-layered structures are higher than the values predicted theoretically based on the 
simplified assumptions. For example, the resonant frequency decreases with increasing resonator 
diameter and alumina substrate thickness for a fixed optimum ratio LID — 0.4 [83]. Fine adjustment 
of resonant frequency is possible by changing air gap thickness between the resonator and the 
metal disk. 

When a dielectric resonator is placed in the vicinity of a microstrip line on the alumina substrate, 
as shown in Figure 8.67(a), magnetic coupling between the resonator and microstrip line is caused, 
increasing if the distance / between the resonator edge and microstrip-line edge decreases [84]. In this 
case, the dielectric resonator operates like a reaction cavity that reflects the radiofrequency energy at 



E field 



H field 



1 

L 

T 



ffft 




TtTT 




DR 






e r =46 






A A AAAA AA ' , 




Ground plane 



FIGURE 8.66 DR-microstrip coupling and electromagnetic fields. 



DIELECTRIC RESONATOR OSCILLATORS 425 




(a) 




FIGURE 8.67 DR-microstrip coupling and equivalent circuit. 



the resonant frequency, which is similar to an open circuit with a voltage maximum at the reference 
plane at the resonant frequency. Because coupling is via the magnetic filed, the dielectric resonator 
appears as a series load on the microstrip line, as shown in the equivalent circuit of Figure 8.67(h). 
The resonator is modeled as a parallel RLC circuit, and the coupling to the line is represented by the 
turns ratio of the equivalent transformer. Since the input impedance Z m of a parallel RLC circuit can 
be written as 



7 



R 



1 + 2jQ- 



. Aw 
co 0 



(8.250) 



where Q — R/(a> 0 L) — cdqCR is the unloaded quality factor of the resonator, co 0 — 1/+/LC is the 
resonant frequency, and Aco — co — a>o, then the equivalent series impedance Z seen by the microstrip 
line is expressed as 



Z = 



N 2 R 



1 + 2jQ- 



Aco 
co 0 



(8.251) 



where N is the turns ratio of the equivalent transformer [85]. 

The coupling factor g between the resonator and the feedline represents the ratio of the unloaded 
Q to external Q e , and can be written as 



g 



Q_ 

2e 



R 



-I 



Rl 



(DqL N 2 C0qL 



N 2 R 
2Z7 



(8.252) 



where R L = 2Z 0 is the load resistance for a feedline with source and termination resistances Z 0 
[86,87]. In some cases, the feedline is terminated with an open-circuit quarterwave line from the 
resonator to maximize the magnetic field at that point, thus resulting in R^ = Zn, and the coupling 
factor is twice the value given in Eq. (8.252). 

The reflection coefficient seen by the source on the terminated microstrip line toward the resonator 
can be written as 



g 
1 + , 



_ (Zn + N 2 R) - Z 0 _ N 2 R 
~ (Z 0 + N 2 R) + Z 0 ~ 2Z 0 + N 2 R 

which can be used to determine the coupling coefficient from the direct measurements as 

i - r 



(8.253) 



(8.254) 



426 OSCILLATORS 



1- 



o 



o 



DR 



DR 



Terminating 
network 



J 



5012 



Terminating 
network 



J 



50 £1 



(a) 



(/>! 



FIGURE 8.68 Schematics of dielectric resonator oscillator. 



which is a strong function of the distance between the resonator and the microstrip line under 
fixed shielding conditions for given substrate thickness and dielectric constant [87,88]. For 60-GHz 
cylindrical dielectric resonators coupled to a microstrip line on a GaAs substrate, it was shown by the 
numerical calculation that the maximum coupling coefficient is provided when the distance between 
the resonator center and the microstrip line is approximately 3/5 of the resonator radius and this 
distance is almost independent of structural parameters [89]. 

A dielectric resonator can be incorporated into the oscillator circuit to improve its frequency 
stability using generally either the parallel feedback configuration shown in Figure 8.68(a) or the series 
feedback arrangement shown in Figure 8.68(b). In a parallel feedback configuration, the dielectric 
resonator is coupled to two microstrip lines, operating as a high-<2 bandpass filter when a portion of 
the transistor output signal is coupled to its input. The amplitude level of coupling is controlled by the 
spacing between the dielectric resonator and the microstrip lines, while the phase is determined by the 
lengths of both microstrip lines. The series feedback configuration with a single microstrip line can be 
easily implemented, but typically does not have a tuning range as wide as that obtained with parallel 
feedback. However, placing the dielectric resonator on the gate port that is isolated from the output 
through the very small drain-gate capacitance minimizes interaction between the oscillator output 
and input circuits, thus resulting in very high loaded quality factors in a series feedback configuration. 
As a result, a series feedback dielectric resonator oscillator (DRO) demonstrates greater capability to 
achieve the lower phase noise level compared to its parallel feedback counterpart [90]. 

Figure 8.69(a) shows the series feedback bipolar oscillator circuit with a dielectric resonator 
(f 0 Q — 10 14 Hz) coupled to the microstrip line, which is included into the base circuit [91]. To 
provide high-power and low-phase noise oscillator performance, a four finger InGaP/GaAs HBT 
device with a total emitter area of 240 txm 2 is used, resulting in a — 124 dBc/Hz at 10 kHz offset from 
a carrier frequency of 6.7 GHz. This HBT DRO together with a 6-dB buffer amplifier was integrated 
on a 0.6-mm alumina substrate. At higher frequencies, the GaAs MMIC technology can be used for 
basic active circuit with external dielectric resonator. Figure 8.69(b) shows such a series feedback 
10.7-GHz DRO with an oscillator chip size of 1.5 x 1.5 mm 2 , where the dielectric resonator (Q — 
7400 and £ r = 36.3) is mounted on alumina substrate and coupled to a microstrip line terminated by 
a 50-Q load [92], The MESFET device with a gate length of 1 (j.m and a width of 300 \im was used 
to obtain an oscillator output power of 10 dBm. The value of a feedback capacitance connected to the 
source terminal is optimized for maximum reflection coefficient at the drain terminal. The resistor and 
quarterwave shunt microstrip line in the source circuit were used for a single power supply operation 
with chosen drain current. By using a 0. 15-jJ.m AlGaAs/InGaAs heterojunction field-effect transistor 
(HJFET) technology, the DRO oscillation frequency can be extended to a V band. In this case, the 
oscillation frequency of 59.6 GHz with an output power of 19 dBm, a phase noise of —90 dBc/Hz at 
100-kHz offset and a temperature stability of 1.6 ppm/°C was achieved using a dielectric resonator 
with an unloaded Q of 5000 and a relative dielectric constant e r = 23.8 [93]. 



DIELECTRIC RESONATOR OSCILLATORS 427 




FIGURE 8.69 Schematics of series feedback dielectric resonator oscillator. 



However, at high microwave frequencies, it is very difficult to effectively manufacture accurate 
high-g dielectric resonators, as well as their physical handling, since a 2-GHz resonator is in the 
order of 1 mm in diameter. Also, at A"«-band frequencies and higher, the gain of the transistors 
for the same output power becomes marginal, and resonator coupling would need to be excessive, 
thus destroying the inherent Q and spectral purity of the oscillator. These problems can be partially 
eliminated by employing a push-push oscillator design approach. Figure 8.70 shows the push-push 
DRO configuration, where outputs from both transistors are combined in parallel using ordinary 
microstrip technique. In this case, each microstrip line has a characteristic impedance of 100 Q and 
is of a half- wavelength at fundamental frequency f 0 , in order to provide a parallel connection of the 
transistors with the output impedances of 100 Q [94]. Since the gate circuits of each MESFET are on 
opposite sides of the resonator, the currents coupled at each gate will be exactly antiphased, as shown 
in Figure 8.66. Under these conditions, each MESFET is phase locked to the other, with their second- 
harmonic power combined in phase at the output of the oscillator and delivered to the 50-f2 load. A 
positive feedback by using a capacitive open-circuit stub is added to each device source. The level of 
the fundamental power at the load is usually of —25 dBc and less, relatively to the second-harmonic 
power, which can be maximized when MESFET is biased near pinch-off or forward conduction. 
The dielectric resonator must be located at the distance of 6 — 180° from the gates, however input 
device reactances should be taken into account, since they provides the effective transmission-line 
electrical length extension. The 50-fi resistors at the ends of the gate transmission lines are necessary 



428 OSCILLATORS 




FIGURE 8.70 Schematic of push-push dielectric resonator oscillator. 



to minimize unwanted parasitic oscillations. The single-sideband phase noise of such a push-push 
DRO at 12 GHz can reach the level of -110 dBc/Hz at 100-kHz offset from the carrier [81]. 



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Phase-Locked Loops 



This chapter begins with introduction of the basic phase-locked loop concept. Then, the basic perfor- 
mance and structures of the analog, charge-pump, and digital phase-locked loops are analyzed. The 
basic loop components such as phase detector, loop filter, frequency divider, and voltage-controlled 
oscillator are discussed, as well as loop dynamic parameters. The possibility and particular realizations 
of the phase modulation using phase-locked loops are presented. Finally, general classes of frequency 
synthesizer techniques such as direct analog synthesis, indirect synthesis, and direct digital synthesis 
are discussed. The proper choice of the synthesizer type is based on the number of frequencies, 
frequency spacing, frequency switching time, noise, spurious level, particular technology, and cost. 

9.1 BASIC LOOP STRUCTURE 

The basic phase-locked loop (PLL) concept has been well known and widely used since it was 
proposed and described by Appleton and later analyzed by Bellescize [1,2]. Starting with the rapid 
development of integrated circuits in the 1970s, the PLL had become an important part of modern 
communication systems, improving their performance and reliability. A PLL circuit responds to both 
the frequency and the phase of the input signal, automatically raising or lowering the frequency 
of a voltage-controlled oscillator until it is synchronized with the reference in both frequency and 
phase. In other words, the PLL controls the phase of its RF output signal in such a way that the 
phase error between output phase and reference phase reduces to a minimum. The basic structure 
of a PLL is shown in Figure 9.1, which consists of a phase detector (PD), a loop filter (LF), and a 
voltage-controlled oscillator (VCO). The PD compares the phase of the input reference signal with 
that of the VCO, and its output voltage is filtered and applied to the VCO whose output frequency 
moves in the direction so as to reduce the phase difference of both input and output signals. As a 
result, when the frequency is locked, the frequency of the VCO is exactly equal to the frequency of 
the reference oscillator. 

To describe the general electrical behavior of the PLL, consider the input reference signal u in and 
output signal v ov n expressed as 

v Jt) = V in cos(« in ? + <9 in ) (9. 1) 

Uout(f) = Vo u tCOS(>W + 0out) (9.2) 

where V m and V 0 m are the amplitudes, oj m and a> oal are the angular frequencies, and 8 m and <p ont are 
the phase constants of the input reference and output VCO signals, respectively [3,4]. If the loop is 
initially unlocked and the PD operating as a signal multiplier has a sinusoidal transfer characteristic, 
the error signal v e at its output is written as 

v e (t) = K d [cos [(&>,„ - w out ) t +(6> in - 0 out )] + cos [(&> in + oj m ) t +(6> in + 4> out )]} (9.3) 



RF and Microwave Transmitter Design, First Edition. Andrei Grebennikov. 
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 



433 



434 PHASE-LOCKED LOOPS 



Reference 
oscillator 


v,„ 


Phase 
detector 


v e 


Loop 
filter 


v c 


Voltage -controlled 
oscillator 



































i\ output 



FIGURE 9.1 Basic structure of phase-locked loop. 



where K a = Vi n V out /2 is the gain factor of the PD. Since the higher frequency component co m + m mt 
is eliminated by the ideal loop low-pass filter (LPF) used, the signal u c at the LPF output can be 
given by 

v c (t) = K a cos [(a) m - <y out ) t +(6» in - 0 out )] . (9.4) 

When the loop is locked after a period of time sufficiently long for transient, the VCO signal v out 
becomes synchronous with the input reference signal y in , and it can be written as 

l>aut(0 = Vout cos(&> in ? + ip out ) (9.5) 

where (p m is the initial phase. Comparing Eqs. (9.2) and (9.5) shows that the phase </> out in Eq. (9.2) 
is a linear function of time expressed by 

<Pom = (ttJin - <«oiit) t + (font (9.6) 

and the LPF output signal v c becomes a dc signal given by 

Vc(t) = K A cos(6> in - <p 0M ) . (9.7) 

The instantaneous angular frequency a} 0ix of the VCO operating as a frequency-modulated oscil- 
lator represents a linear function of the control signal ii c around the center angular frequency w out 
according to 

d 

Wosc = — (<*W + <^out) = «out + K wco v c (t) (9.8) 
at 

and 

d<p 0 - 



dt 



= K wco v c (t) (9.9) 



where K\co is the VCO sensitivity measured in radian per second per volt and (o olJt is the frequency 
that occurs when the PD is in the center of its output transfer characteristic or the frequency midway 
between the frequencies that occur at the edges of the PD range. 
From Eqs. (9.6), (9.7), and (9.9) it follows that 

£»ta - Wout = K A K VC o cos(0 in - <p out ) (9.10) 

resulting in 



(9.11) 



ANALOG PHASE-LOCKED LOOPS 



435 



Substituting Eq. (9.1 1) into Eq. (9.7) yields 

v c = — (9.12) 

Avco 

which clearly shows that it is the dc signal v c that changes the VCO frequency from its central value 
o>out to the input signal angular frequency &>;„ according to 

Wosc = Wout + ^VCOUc = Win- (9.13) 

If the angular frequency difference co m — co oul is much lower than the loop gain K — K A Kv C o, 
Eq. (9.11) is simplified to 6 m — <p out « nil, thus indicating that the VCO output signal is almost in 
phase quadrature with the input reference signal when the loop is locked, where the phase quadrature 
condition corresponds to a> m — a> 0llt . In this case, by letting 6 0ut — <p out + nil, Eq. (9.7) can be 
rewritten as 

v c (t) = K d sm(6 m -6 0ut ) (9.14) 

where 6 e — 0 m — 6 0ut is the phase error between the two signals, which becomes zero when the 
initial frequency offset equals to zero. It should be noted that Eq. (9.14) takes into account the phase 
shift of 180° around the loop. For a sufficiently small phase difference 6 e — (9 in — 6 0ut , the following 
approximation can be used 

u c (0 w KMn - flout) (9.15) 

with the PD gain factored measured in volt per radian. When the difference \a> ia — co ollt | exceeds the 
loop gain K in a PD with sinusoidal transfer characteristic, synchronization can no longer maintain 
according to Eq. (9.1 1), and the loop falls out of lock. 

As one of the first practical implementations, the PLL structure was used to stabilize the operating 
frequency of a power ultra-high-frequency (UHF) VCO developed as part of a solid-state microwave 
radio relay system [5]. In this case, in order to increase the reference frequency provided by a crystal 
oscillator, the frequency doublet' and triplet' with buffer amplifiers were used. Integrated PLLs are 
now well developed and widely used in a variety of applications such as a reference source, signal 
modulator, frequency-selective demodulator, or frequency synthesizer. 



9.2 ANALOG PHASE-LOCKED LOOPS 

The basic analog PLL structure also includes a frequency divider (DIV) between the oscillator and the 
PD to match the frequency of the reference signal, as shown in Figure 9.2(a), since it creates enough 
flexibility in designing of both reference oscillator and VCO with improved performance. When the 
system is phased-lock, the VCO is N times that of the reference frequency, where TV is the frequency 
division ratio. Changing the value of the divider will cause the PD to sense a frequency error. In this 
case, the feedback will respond with a correcting voltage. The operating range is set by the maximum 
frequency of the divider, the division ratio of the divider, and the VCO tuning range. The PD, LF, 
and VCO compose the feedforward path with the feedback path containing the DIV. Removal of the 
DIV produces unity gain in the feedback path for N — 1 , and the VCO output frequency is equal to 
the PD input frequency. 

An analog PLL is generally a nonlinear system since its PD is described by a nonlinear transfer 
characteristic in a common case. However, when its loop is locked, a PLL can be accurately ap- 
proximated by a linear model when the phase error 8 t is small and analyzed using the concepts and 
terminology of feedback control systems based on Laplace transform. In this case, each component 



436 PHASE-LOCKED LOOPS 



Reference 
oscillator 



Phase 
detector 



Loop 
lllter 



Voltage -controlled 
oscillator 



rrequenc) 
di\ idei 



RF output 
signal 



(a) 



0,„(.v) 




k. 



(6) 



FIGURE 9.2 Typical structure of analog phase-locked loop. 



of the PLL can be described by a linear transfer function. When the loop is closed, the response of 
any component and entire system can be expressed as the ratio of phases at corresponding output 
and input points. Figure 9.2(b) shows the typical structure of a linearized analog PLL where the PD 
is assumed to be sinusoidal, operating linearly over a phase range of ±1 rad and represented by the 
frequency multiplier with separate gain block, F(s) is the filter transfer function, and s — ja> is the 
complex frequency (Laplace) variable. The VCO is usually modeled by an integrator, since phase is 
the integral of instantaneous frequency with Ky C o- 

According to feedback control system analogy, the system forward gain is written as 

K d K VC oF(s) 

G(s) = (9.16) 

with the open-loop transfer function or open-loop gain GH(s) — G(s)/N, whereas the closed-loop 
transfer function H(s), which is a measure of the loop response to changes in the input phase or 
frequency, is obtained by 

9'(s) G(s) K d KvcoF(s) 

9 m (s) l+G(s)/N s + K d (K vco /N)F( S y 

where the roots of 1 + G(s)/N are the poles of the system function, determining the transient behavior 
of the loop. This affects the ability of the loop to follow rapid changes in input frequency and phase. 

By considering the change in output frequency produced by introducing a test frequency at various 
point in the PLL, all transfer functions from these points to output can be derived, which are shown 
in Table 9.1 [6,7]. Note that the transfer function of the LF is a major factor in loop performance. 
As the filter bandwidth is reduced, its response time is increased. This helps to keep the loop in lock 
through momentary losses of input signal, and minimizes the noise transmitted through the loop at 
the expense of a reduction in the capture range. In this case, the phase error function is written as 

s6 in (s) 
i + KiKycoFisJ/N 



ANALOG PHASE-LOCKED LOOPS 437 

TABLE 9.1 Transfer Functions for Various Analog PLL Ports. 

Source to Output Transfer Function 

G(s) 

Reference oscillator 



N divider 
Phase detector 
Loop filter 
VCO 



1 + G(s)/N 

G(s) 
1 + G(s)/N 

1 gw 

K d 1 + G(s)/N 

K\co 1 
i 1 + G(s)/N 

1 



1 + G(s)/N 



The order of the loop equals the number of poles in the open-loop transfer function GH(s) defined 
by the highest power of s in its denominator [8]. Therefore, a PLL without LF when F(s) — 1 is called 
the first-order PLL, because the highest power of .? in the denominator of G(.v) is 1. In this case, the 
open-loop gain of a first-order PLL is defined by the forward gain and equals to K^K^co^N)/.^, where 
K — K^KycolN represents a dc loop gain referred to the PD. This is a type 1 system when the phase 
error function is simplified to 

0 e {s)= '1 ... (9.19) 

s + K d K WC0 /N 

where the dc loop gain is defined according to Figure 9.1 by taking into account Eqs. (9.12) and 
(9.15) as 

^ v e Aa> Acq 

6 e v e 6 e 

Equation (9.17), which can simply be rewritten as H(s) — 1/(1 + s/A')forA'= 1 , represents an LPF 
characteristic with a cutoff (—3 dB) frequency of co — K. Equation (9.18), rewritten for 6 e (s)l9- m {s) 
as sl{s + K) for N = 1, shows that the relative error phase is characterized by a high-pass filter 
characteristic. 

The steady-state phase error resulting from a step change of input phase of magnitude A6 in when 
0 to (i) = A8 in /s is derived in a stable feedback control system according to 

A9 e (s) = lims<9 e (s) = lim * A ° m — = 0 (9.21) 

s _»o s ->o s + K d Kvco/N 

and the steady-state error resulting from a ramp input phase, or from a step change in reference 
frequency of magnitude Aa> when 6> in (.v) = Awls 2 is equal to 

Aw AwN 
AO e (s) = lim jfl e (j) = lim — — — — = — (9.22) 

s _>0 S ^0 S + K d Kyco/N KiKyco 

which is a constant [6]. In addition, it is important to know the first-order PLL response to a ramp 
change in frequency with time at a rate of dAcoldt when 9 m (s) — (1/s 3 ) dAcoldt. As a result, 

dAco/dt 

A0 e (j) = lim L = oo (9.23) 

s 2 + sK d Kvco/N 



438 PHASE-LOCKED LOOPS 



which means that above some critical value of change rate of reference frequency the loop will no 
longer stay locked or, conversely, if VCO frequency is linearly swept at a rate above a critical value 
to achieve locking, the latter will not occur. 

These results indicate that a first-order PLL will eventually track out any step change in input 
phase that is within the system hold-in range and will follow a step frequenc